mcla concise review

8/11/2019 MCLA Concise Review

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MULTIVARIABLE CALCULUS AND LINEAR ALGEBRA CONCISE REVIEW

VARUN DANDA AND SIDDHARTH KURELLA

Part 1. Multivariable Calculus

1. Parametric Equations

Supposex fptq and y gptqthen the variable t is considered a parameterand the equations are called parametric equations

As t varies, the pointpx, yq varies and traces out a curve that is called a parametric curve

Theslope of a parametric curve is

dy

dx dy{dt

dx{dt (1)

Proof:

dy

dt dy

dxdx

dt (Chain Rule)

Given xptq and yptq

A

b

a

y dx

yp

tq

x1

ptq

dt (2)

where t increases from to

The lengthof a curve in the form fpxq is defined as

L ba

d1 `

dy

dx

2

dx (3)

Substituting equation1 into2, one can find the equation for the length of a parametric curve

L

ddxdt

2 ` dydt

2dt (4)

Thesurface area of a parametric curve rotated around the x or y axis is

x axis: S

2y ds ; y axis: S

2xds (5)

where ds c

dx

dt

2

`dy

dt

2

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2. Polar Coordinates

TheCartesian coordinatespx, yq of a polar coordinate (r, q is defined byx r cos ; y r sin (6)

Therefore : r2 x2 ` y2 and tanpq yx

Figure 1. Polar Coordinate Representation

The graph of a polar equation, r

f

p

qconsists of all points P that have at least one polar

representationpr, q whose coordinates satisfy the equationTo find atangentline to a polar curve r fpq, is regarded as a parameterand the parametric

equations are written:

x r cospq fpq cospthetaq and y r sinpq fpq sinpthetaqTo find slope use equation 1 in terms of polar coordinates

dy

dx dy{d

dx{ddrd

sinpq ` r cospqdrd

cospq r sinpq (6)

The area of a sectorof a circle isA 1

2r2 (7)

Considering infinitesimally small sectors of circles, the area of a polar curve between two angles is

A ba

1

2r2 d (8)

To find the length of a polar curve simplify`dxdt

2 ` `dy

dt

2

dx

d dr

dcospq r sinpq ; dy

d dr

dsinpq ` r cospq

dx

d

2 `

dy

d

2

dr

dcospq r sinpq

2 `

dr

dsinpq ` r cospq

2

r2 `

dr

d

2

Therefore, plugging into equation 4,

L

dr2 `

dr

d

2

dt (9)

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3. Sequences and Series

A sequencecan be thought of as a list of numbers written in a definite order

If a sequence approaches a certain value it is considered convergent, otherwise it is divergent

A sequence is consideredincreasingif every term after the first is greater than the preceding terms

anddecreasing if every term after the first is lower than the preceding termsA sequence is consideredmonotonic if it is either increasing or decreasing

A sequence isbounded above if there exists a number M such that an M for all n 1A sequence is bounded below if there exists a number M such that an M for all n 1

Theorem: Every bounded, monotonic sequence is convergent

Aseries is the addition of the terms of an infinite sequence and is denoted by

8

n1

an a1 ` a2 ` a3 ` ` an `

Theorem: If limn8 an exists or =0 then the series8

n1 an is convergent, otherwise it is divergent

The Integral Test: If f is a continuous, positive, decreasing function onr1, 8q and let an fpnq,then the series

8n1 an is convergent if and only if the improper integral

81

fpxq dxis convergent

P-Series:8

n11

np is convergent ifp 1 and divergent ifp 1;Proved by integral test

The Comparison Test: Suppose that

an and

bn are series with positive terms(i) If

bn is convergent and an bn for all n, then

an is also convergent

(ii) If bn is divergent and an bn for all n, then an is also divergentAnalternating series is a series whose terms are alternately positive and negative

Example:

8n1

p1qn n

If the alternating series8

n1 p1qn1 bn satisfies (i) bn`1 bn (ii) limn8 bn 0Then the alternating series is convergent

A series an is absolutely convergentif the series of the absolute values |an

|is convergent

and is called conditionally convergent if it is convergent but not absolutely convergent

The Ratio Test:

limn8

an`1an

L

(i) IfL 1 then the series8n1 an is absolutely convergent and therefore convergent(ii) IfL 1 or L 8 then the series8n1 an is divergent(iii) IfL 1 then the ratio test is inconclusive

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The Root Test:

limn8n

a|an| L

(i) IfL 1 then the series8n1 an is absolutely convergent and therefore convergent(ii) IfL 1 or L 8 then the series8n1 an is divergent(iii) IfL 1 then the ratio test is inconclusive

Apower series is a series of the form8

n0cn px aqn c0 ` c1 px aq ` c2 px aq2 `

Theorem: For a power series8

n0 cnpx aqn there areonly three options:(i) The series converges only when x a(ii) The series converges for all x(iii) There is a positive number R, called the radius of convergence, such that the seriesconverges if|x a| Rand diverges if|x a| R

Theinterval of convergence of a power series is the interval that consists of all values of x forwhich the series converges

ATaylor Series centered at a is given by:

fpxq 8

n0

fpnqpaqn!

px aqn wherecn fpnqpaq

n! in the power series form

The special case where a=0 is called a Maclaurin Series

Important Maclaurin Series:

1

1 x8

n0xn

ex 8

n0

xn

n!

sinpxq 8

n0p1qn

x2n`1

p2n ` 1q!

cospxq 8

n0p1qn x

2n

p2nq!

tan1 x 8

n0p1qn x

2n`1

p2n ` 1q

Above series were derived by differentiating and integrating other series

4. Vectors

Avector is a quantity that has both a magnitude and a direction

Vector Addition: To add two vectors geometrically, align the head of the first vector beingadded to the tailof the second and draw a ling from the tail of the first to the head of the second

Scalar Multiplication: The scalar multiple of a vector is cv whose length is|c| times the lengthofv and whose directions is the same as v ifc 0 and opposite ifc 0

Themagnitude of a vector is the length of it and is represented by either|v| or||v||; In terms ofitscomponents, the magnitude of a vector is|a|

aa21` a2

2` ` a2n where n is the number of

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dimensions the vector is in

Vectors can also be added algebraicallyby adding their components; For example, ifu ru1, u2sand v rv1, v2s then u ` v ru1 ` v1, u2 ` v2s

Thestandard basis vectors i, j, k can be used to express any vector in V3

i x1, 0, 0y;j x0, 1, 0y; k x0, 0, 1yFor example a xa1, a2, a3y xa1, 0, 0y ` x0, a2, 0y ` x0, 0, a3y a1i ` a2j ` a3k

Aunit vector is a vector whose length is 1 such as i, j, and k; The unit vector that has the samedirection as another vector, a is u 1|a|a

Thedot product of two vectors a xa1, a2, a3y and b xb1, b2, b3yis given bya b a1 b1 ` a2 b2 ` a3 b3 which gives a scalar

The dot product can also be expressed by using the law of cosines

a b |a| |b| cospq(10)Theorem: From the above definition: Two vectors a andb are orthogonal if and only ifa b 0

The scalar projectionandvector projection of a vector b onto another vectora is defined as

compab a b|a| and projab a b|a|2 a (11)

Thecross productof two vectors a xa1, a2, a3yandb xb1, b2, b3y is given bya b xa2b3 a3b2, a3b1 a1b3, a1b2 a2b1ywhich results a vector

An easier way to view the cross product is to use determinants:

a b

i j ka1 a2 a3b1 b2 b3

(12)

The cross product is also expressed as

|a b| |a||b| sinpq (13)The length of the dot producta bis equal to the area of the parallelogram determined by a and b

Properties of Dot Products: Supposea,b,andc are vectors and c is a scalar

a a |a|2a pb ` cq a b ` a c

a b b apcaq b cpa bq a pcbq

0 a 0

Properties of Cross Products: Supposea, b,andcare vectors and c is a scalar

a b b apcaq b cpa bq a pcbq

a pb ` cq a b ` a ca pb cq pa bq c

a pb cq pa cqb pa bqc5


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5. Vector Functions

Avector function is a function whose domain is a set of real-numbers and range is a set of vectors:

rptq xfptq, gptq, hptqy fptqi ` gptqj ` hptqkThe components of vector function can be considered parametric equations that can be used to

sketch the curve

The derivative of a vector function is defined as follows:

dr

dt r1ptq xf1ptq, g1ptq, h1ptqy f1ptq i ` g1ptqj ` h1ptq k

The integral of a vector function is defined as follows:ba

rptq dt b

a

fptqdt

i `b

a

gptqdt j `b

a

hptqdt

k

The length of a vector function can be determined by using equation 4 in three dimensions

L ba

arf1ptqs2 ` rg1ptqs2 ` rh1ptqs2s dt whererptq xfptq, gptq, hptqyor the more compact form: L

ba

|r1ptq| dt (14)

Theunit tangent vector indicates the direction of the curve at a certain point and is given by

Tptq r1ptq

|r1ptq| (15)

Thecurvature

of a curve at a given point is a measure of how quickly the curve changes at thatpoint

dTds

(16)

It is, however, easier to compute curvature in terms of the parameter t instead of s

dT

dt dT

ds

ds

dt and therefore

dT{dtds{dt

So

pt

q |T1ptq||r1ptq|

(17)

Theunit normal vector is orthogonal to the unit tangent vector and points in the direction thatthe curve is curving towards

Nptq T1ptq

|T1ptq| (18)

Thebinormal vectorcan be found by using the right hand rule on the tangent and normal vectors

Bptq Tptq Nptq (19)6


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Figure 2. Tangent, Normal, and Binormal Vectors

6. Cylindrical and Spherical Coordinates

Apart from the commonly used Cartesian Coordinates Cylindrical Coordinates andSpherical Coordinates are used in special cases to easily solve certain problems.

In a Cylindrical Coordinate system, a point P is represented bypr,,z q, where r and are the polarcoordinates of the projection of P onto the xy -plane and z is the distance from the point to the

xy-plane.

Using geometry, Conversions from cylindrical to cartesian coordinates and vice versa are below:

x r cos ; y r sin ; z z | r2

x2

` y2

; tan y

x ; z zIn a Spherical Coordinate system, a point P is represented byp ,,q, where is the distance from

the origin to point P, is the angle that the projection of P makes with the x-axis, and is theangle that the line from the origin to P makes with the z-axis; 0 and 0

(a) 2 dimension Vector Field (b) 3 dimension Vector Field

Figure 3. Vector Fields

The conversions from spherical to cartesian coordinates and vice versa are below:

x sin cos ; y sin sin ; z cos | 2 x2 ` y2 ` z2

7. Partial Derivatives

Suppose fis a function of two variables, xand y , and we let only xvary while keeping y fixed at aconstant. Then we are really considering a function of a single variable x, namely, gpxq fpx, bq. If

g has a derivative at a, then we call it the partial derivative of fwith respect to x atpa, bqTherefore:

fxpa, bq g1paq where gpxq fpx, bqPartial Derivative Notations:

fxpx, yq fxBfBx Dxf7


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fypx, yq fyBfBy DyfTo find partial derivatives with respect to a variable consider the other variables as constants and

differentiate the function with respect with the variable. Note: Partial Derivatives can be found forfunctions with more than 2 variables.

Higher derivativescan also be applied to partial derivatives:

pfxqx fxx BBxBf

Bx

B2f

Bx 2

pfxqy fxy BByBf

Bx

B2f

ByBx

pfyqx fxy BBxBf

By

B2f

BxBy

pfyqy fyy BByBfBy B

2

fBy 2Partial Derivatives can help to compute the tangent plane to a surface that can be used to

perform linear approximations

The equation of a tangent plane to the surface z fpx, yq at the point Ppx0, y0, z0q isz z0 fxpx0, y0qpx x0q ` fypx0qpy0qpy y0q

This can be used to find approximate values of points near the one used for the tangent planeapproximation

Thechain rule can also be applied to derivatives of functions with more than one variable.Suppose that z fpx, yq is differentiable and x gptq and y hptq. Then:

dz

dtBfBx

dx

dt`BfBy

dy

dt (20)

Ifx gps, tq and y hps, tq then:BzBs

BzBx

BxBs`

BzBy

ByBs and

BzBt

BzBx

BxBt`

BzBy

ByBt (21)

Example: Write out the Chain Rule for the case where w

f

px , y , z , t

qand

x xpu, vq, y ypu, vq, z zpu, vq, and t tpu, vqBwBu

BwBx

BxBu`

BwBy

ByBu`

BwBz

BzBu`

BwBt

BtBu

BwBv

BwBx

BxBv`

BwBy

ByBv`

BwBz

BzBv`

BwBt

BtBv

The directional derivative of f atpx0, y0q in the direction of a unit vectoru xa, by is

Dufpx0, y0q limh0 fpx0 ` ha, y0 ` hbq fpx0, y0qh

(22)

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Ifu ithen Dif fx and ifu j then Djf fy Therefore the partial derivatives of f withrespect to x and y are special cases of the directional derivative An easier way to represent the

directional derivative in the direction of any unit vector u xa, by isDufpx, yq fxpx, yqa ` fypx, yqbor more simply Dufpx, yq xfxpx, yq, fypx, yqy u

The termxfxpx, yq, fypx, yqy is also used commonly and is called the gradient vector and is denoted byfpx,

The gradient vector represents the direction of fastest increase of fIt is easy to see that Dufpx, yq fpx, yq u (23)

The directional deriviative and gradient can be applied in three dimensions as well:

fBfBx i `BfByj `

BfBzk and Dufpx,y,zq fpx,y,zq u

Just as with single variable functions, themaximaand minimaof multivariable functions can alsobe found

Theorem: Iffhas a local maximum or minimum at (a,b) and the first-order partial derivativesexist there, then fxpa, bq 0 and fypa, bq 0. The points at which fxpa, bq 0 and fypa, bq 0 are

calledcritical pointsSecond Derivatives Test: Suppose (a,b) is a critical point.

Let D

fxx fxyfyx fyy

fxx fyy pfxyq2

(a)IfD 0 and fxxpa, bq 0 then fpa, bq is alocal minimum(b) IfD 0 and fxxpa, bq 0 then fpa, bq is a local maximum(c) IfD 0 then fpa, bq is neither a local minimum nor maximum but a saddle point

8. Multiple Integrals

The concept of an integral can be applied to multiple dimensions. Double Integrals are integralsof two-variable functions that represent the volume under this curve. Riemann sums can be used to

estimate these integrals by adding up rectangular prisms instead of rectangles.

Figure 4. Riemann Sum of a 3-Dimensional function

Double integrals can be solved one integral at a time as an iterated integraldc

ba

fpx, yqdxdydc

ba

fpx, yqdx

dy

Theorem: If the rectangle R px, yq|a x b, c y d, thenR

fpx, yq dA dc

ba

fpx, yq dxdyba

dc

fpx, yq dy dx

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Theorem:

R

gpxq hpyq dA ba

gpxq dxdc

hpyq dy

whereR ra, bs rc, ds, a rectangleAlthough, these calculations may seem simple, regions that are being integrated over are not always

rectangles q A plane region D is said to be Type Iif it lies between graphs of two continuousfunctions of x. Similarly it is said to be Type II if it lies between two continuous functions of y.

Figure 5. Type I and Type II Regions

Iffis continuous on a type I region D such that D tpx, yq|a x b, g1pxq x g2pxqusq

then

D

fpx, yq dA dc

g2pxqg1pxq

fpx, yq dy dx (24)

Iff is continuous on a type II region D such that D tpx, yq|c x d, h1pyq x h2pyqusq

then

D

fpx, yq dA dc

h2pyqh1pyq

fpx, yq dxdy (24)

Example: EvaluateD px ` 2yq dAwhere D is the region bounded by the parabolas y 2x

2

andy 1 ` x2

From the graph the region D can be described as

D tpx, yq| 1 x 1, 2x2 y 1 ` x2u

Figure 6. y 2x2 and y 1`x2

Therfore

D

px ` 2yqdA

1

1

1`x2

2x2px ` 2yqdydx

1

1rxy ` y2sy1`x2

y2x2 dx 32

15

Therefore the Volume formed by z x ` 2y over region D is 3215

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Triple Integralsare used similarly to double integrals

E

fpx,y,zqdVD

u2px,yqu1px,yp

fpx,y,zqdz

dA

ApDq D

dAand VpEq E

dV

Multiple integrals can also be applied to regions defined by polar, cylindrical, and sphericalcoordinates

Iff is continuous on a polar regions of the form D tpr, q| , h1pq r h2pqu

Then

D

fpx, yq dA

h2pqh1pq

fpr cospq, r sinpq r d r d (25)

Figure 7. D tpr, q| , h1pq r h2pqu

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Example: Use a double integral to find the are enclosed by one loop of the four leaved roser cosp2q

Figure 8. r cosp2q

ApDq D

dA {4{4

cosp2q

0

r d r d {4{4

1

2r2cosp2q

0

d 8

The formula for triple integration in cylindrical coordinates:E

fpx,y,zq dV

h2pqh1pq

u2pr cospq,r sinpqqu1pr cospq,r sinpqq

fpr cospq, r sinpq rdrd (26)

Figure 9. Triple Integral of Cylindrical Coordinates

Figure 10

Example: Evaluate 2

2

?4x2

?4

x2

2

?x2

`y2

px2 ` y2q dz dy dx

This iterated integral is a triple over the solid region:

E tpx,y,zq| 2 x 2, ?

4 x2 ya

4 y2,a

2 ` y2 z 2uUsing cylindrical coordinates and the fact that r

ax2 ` y2 the solid region can be represented as:

E tpr,,z q|0 2, r 2, r z 2uTherfore

2

2

?4x2

?4x2

2

?x2`y2

px2 ` y2q dz dy dx

2

0

2

0

2

r

r2 r d z d r d 165

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The formula for triple integration in spherical coordinates:E

fpx,y,zq dVdc

ba

p sin cos , sin sin , cos q 2 sin d dd (27)

Where E is given by E tp ,,q|a b, , c du

Figure 11. Spherical Volume Element: 2 sin d d d

When using spherical coordinates for triple integrals use the fact that a

x2 ` y2 ` z2 tosimplify the integral

To determine limits for triple integrals in spherical or cylindrical coordinates use a process similarto the one below:

Figure 12. Process for Triple Integrals in Spherical Coordinates

9. Change of Variables in Multiple Integrals

Consider the displacement vectorrpx, yq and the new set of variables u and v where x xpu, vq andy ypu, vq.

R

fpx, yq dxdyS

fpxpu, vq, ypu, vqqBpx, yqBpu, vq

dudv (28)

WhereBpx, yqBpu, vq

BxBu

BxBv

ByBu

ByBv

ffand is called the Jacobian

Proof:

d A BrBudu BrBvdv

BrBuBrBvdudv

In cartesian coordinates d A dx ` dy

Thereforedxdy

BrBu

BrBv

dudv where

Bpx, yqBpu, vq

BrBu

BrBv

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Similarly in three dimensions:R

fpx, yq dxdyS

fpxpu,v,wq, ypu,v,wq, zpu,v,wqq Bpx,y,zqBpu,v,wq

dudv (29)

WhereBpx,y,zqBpu,v,wq

BxBu

BxBv

BxBw

ByBu

ByBv

ByBwBz

BuBzBv

BzBw

fiffiffifl

Example: Use formula 29 to derive the formula for triple integration in spherical coordinates.

x sin cos y sin sin z cos

Bpx,y,zqBp ,,q

BxB

BxB

BxB

ByB

ByB

ByB

BzB

BzB

BzB

=

sin cos sin sin cos cos sin sin sin cos cos sin

cos 0 sin

=2 sin

10. Vector Calculus

AVector Field on two-dimensional space is a function Fthat assigns each pointpx, yq with a twodimensional vectorFpx, yq

F can be written in terms of its component functions P and Q as follows:

Fpx, y

q P

px, y

qi`

Qp

x, yqj (30)

Similarly for three dimensional vector fields:

Fpx,y,zq Ppx,y,zqi ` Qpx,y,zqj ` Rpx,y,zqk (31)

(a) 2 dimension Vector Field (b) 3 dimension Vector Field

Figure 13. Vector Fields

It is important to note that the gradient vector is also a vector field called thegradient vectorfield

fpx, yq fxpx, yqi ` fypx, yqjIf f is defined on a smooth curve C with two dimensions then the line integral of f along C is

c

fpx, yq dswhere ds c

dx

dt

2

`dy

dt

2

(32)

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Line integrals can also be respect to x and y:C

fpx, yq dx ba

fpxptq, yptqq x1ptq dt (33)C

fpx, yq dyba

fpxptq, yptqq y1ptq dt (34)Suppose that the curve Cis given by a vector function r

pt

q, a

t

b. Then:

C

F dr ba

Fprptqq r1ptq dt C

F Tds (35)Line integrals are treated similarly in three-dimensional space

(a) (b) (c)

(d) (e) (f)

Figure 14. Line Integral

C

F dr C

P dx ` Q dy|R dz

Theorem: Let Cbe a smooth curve given by the vector function rptq, a t b.ThenC

f dr fprpbqq fprpaqq (36)Proof:

C

f dr ba

Fprptqq r1ptq

ba

BfBx

BxBt`

BfBy

ByBt`

BfBz

BzBt

dt

b

a

d

dtfprptqq dt( Chain Rule) fprpbqq fprpaqq (Fundamental Theorem of Calculus)

Theorem:C

F dris independent of pathin D if and only ifC

F dr 0 for every closed pathC in D.

Theorem: IfC

F dris independent of path in D, the Fis a conservative vector field on D;that is, there exists a function f such that f F.

Theorem: IfFpx, yq Ppx, yqi ` Qpx, yqj is a conservative vector field then:BPBy

BQBx (37)

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Proof:

F fPi ` Qj BfBx i `

BfByj

ThereforeP

Bf

Bx and Q

Bf

ByBPBy

B2fByBx

B2fBxBy

BQBx (Clairauts Theorem)

Greens Theorem: Let Cbe a positively oriented, simple closed curve in the plane and let D be theregion bounded by C.

C

P dx ` Q dyD

BQBx

BPBy

dA (38)

IfF Pi ` j ` R kthen thecurlofF is

F curl F

i j k

BBx

BBy

BBz

P Q R

Theorem: IfFis a vector field and curl F 0 then F is a conservative vector field

IfF Pi ` j ` R kthen thedivergence ofF is

div F F BPx

` QBy`BRBz

Acts On GivesGradient Scalars VectorsDivergence Vectors ScalarsCurl Vectors Vectors

Similar to line integrals,Surface Integrals can be thought of as the volume of the space carvedout by a surface over a space

S

fpx,y,zq dSD

fpx,y,gpx, yqqd

BzBx2

`Bz

Bx2

` 1 dA D

fprpu, vqq |ru rv| dA (39)

The surface integral over a continuous vector field F is called theflux ofFover SS

F dSD

F pru rvq dA

Stokes Theorem: LetSbe a positively oriented, simple closed surface that is bounded by a curve Cwith positice orientation. Let Fbe a vector field whose components have continuous partial

derivatives. ThenC

F dr S

curl F dS (40)

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Proof: F dr

P dx ` Q dy ` R dz

BP

ByBy Bx `BPBzBzBx `

BQBxBx By `

BQBzBzBy `

BRBxBx Bz`

BRByBy Bz

BPBypBAkq `BPBzpBAjq `BQBxpBAkq `BQBzpBAiq `BRBxpBAjq `BRBypBAiqBP

BzBRBx

BAk `

BQBx

BPBy

BAj `

BRBy

BQBx

BAi

curl F dA

Example: Use Stokes Theorem to evaluateC

, F drwhere Fpx,y,zq y2i ` xj ` z2k and C isthe curve of intersection of the plane y ` z 4 and the cylinder x2 ` y2 a2

Figure 15. Intersection ofy `z 4 and x2 `y2 a2

curl F

i j kBBx

BBy

BBz

y2 x z2

p1 ` 2yq k

C

F dr S

curl F dS D

p1 ` 2yq dA

20

10

p1 ` 2r sin q r d r d

Divergence Theorem: Let Ebe a simple solid region and let Sbe the boundary surface of E, givenwith positive orientation. LetF be a vector field whose component functions have continuous

partial derivatives. ThenS

F dS E

div F dV (41)

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Proof: F dS

Fi dSi `

FjdSj `

FkdSk

BFi BSi `

BFj BSj `

BFk BSk

BFi

Bx BxByBzBFj

ByByBzBx `BFk

BzBz BxByBFx

Bx `BFyBy `

BFzBz

BV

p Fq dV

Example: Find the flux of the vector field F zi ` y j ` x k through the spherical surface of radiusa centered at the origin.

div F Bz

Bx`By

By`Bx

Bz 0 ` 1 ` 0 1S

F dS B

div F dVB

1 dV VpBq 43

r3 43

a3

11. Second-Order Differential Equations

ASecond-order linear differential equation has the form

Ppxqd2

ydx2

` Qpxqdydx

` Rpxqy Gpxq (42)

The cases where Gpxq 0 are called homogenous linear equations and the cases where Gpxq 0are called non-homogeneous linear equations

Theorem: Ify1pxq and y2pxq are both solutions of the linear homogeneous equation 42and c1 andc2 are any constants, then the function

ypxq c1y1pxq ` c2y2pxqis also a solution of Equation 42

Proof:Ppxqy2

1` Qpxqy1

1` Rpxqy1 0 Ppxqy22 ` Qpxqy12 ` Rpxqy2 0

Ppxqy2 ` Qpxqy1 ` Rpxqy Ppxqpc1y1 ` c2y2q2 ` Qpxqpc1y1 ` c2y2q1 ` Rpxqpc1y1 ` c2y2q Ppxqpc1y21 ` c2y22q ` Qpxqpc1y11 ` c2y12q ` Rpxqpc1y1 ` c2y2q c1rPpxqy21 ` Qpxqy11 ` Rpxqy1s ` c2rPpxqy22 ` Qpxqy12 ` Rpxqy2s c1p0q ` c2p0q 0

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The term ypxq c1y1pxq ` c2y2pxq is also called the general solutionIf the second-order linear equation has constants for Ppxq, Qpxq, Rpxq then theauxillary

equation(orcharacteristic equation) is

ar2 ` br ` c 0 where ay2 ` by1 ` cy 0The roots of this equation, r1 and r2, are used to find the general solution of the general equation

Case 1: b2 4ac 0y c1er1x ` c2er2x

Case 2: b2 4ac 0y c1erx ` c2xerx

Case 3: b2 4ac 0y expc1cos x ` c2sin xq

On the other hand, the general solution of a nonhomogeneous differential equation can be written as

ypxq yppxq ` ycpxqwhereyp is aparticular solution and yc is the general solution of the homogeneous form

of the equation

Procedure to find the particular solution: Substitute yppxq a polynomial of the same degree as Ginto the differential equation and determine the coefficients. Example: Solve y2 ` 4y e3x

yc c1cos 2x ` c2sin 2x(Using method to find homogeneous solutionFor the particular solution we try yp Ae3x. Then y 1p 3Ae3x and y2p 9Ae3x

Therefore 9Ae3x ` 4pAe3xq e3xso 13Ae3x e3x and A 113

yppxq 113

e3x

The general solution is thus y

yc`

yp

ypxq 113

e3x ` c1cos 2x ` c2sin 2xIf initial conditions were provided, the constants c1 and c2 can be found.

Part 2. Linear Algebra

12. Linear Combinations

Alinear combination is when a vector is equal to the sum of a scalar multiple of other vectors.

In mathematics, this is:v c1v1 ` c2v2 ` c3v3 ` . . .

The scalars c1, c2, and c3, etc. are called thecoefficientsof the linear combination.Using linear combinations, one can create a new coordinate grid expressing all points as linear

combinations of two initial vectors.

13. Vector Inequalities

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TheTriangle Inequalitystates the following:For all vectors u and v in Rn:

u ` v u ` vProof:

This inequality will be proven by proving thesquare of the left hand side is less than or equal

to the square of the right hand side.u ` v2 pu ` vq pu ` vq

u u ` 2pu vq ` v v u2 ` 2|u v| ` v2 (Absolute Value is non-negative) u2 ` 2uv ` v2 (Cauchy-Schwarz Inequality) pu` vq2

Since the squares of both sides satisfy theinequality, and the sides are non-negative, the

inequality must be true.

TheCauchy-Schwarz Inequalitystates thefollowing:

For all vectors u and v in Rn:

|u v| uvProof:

|u

v|

uv|cos |

Since|cos | 1, the inequality is true, asuv|cos | uv

14. Lines and Planes

In Rn, thenormal form of the equation for a line is:

n px pq 0 orn x n pwherep is a specific point on the line and n, which is not0, is a normal vector for the line .

Thegeneral form for the equation of a line is:

ax ` by c

where a normal vector n for the line is equal to

ab

.

Thevector form of the equation of a line in R2 or R3 is:

x p ` tdwherep is a specific point on the line andd, which is not 0, is a direction vector for the line .

By taking each component of the vectors, the parametric equations of a line are obtained.

Thenormal form of the equation of a plane in R3 is:

n px pq 0 orn x n pwherep is a specific point on P andn, which is not0, is a normal vector for P.

Thevector form of the equation of a plane in R3 is:

x p ` su ` tvwherep is a point on P andu and v are direction vectors for the plane P(and are non-zero,

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The equations that result from each component of the vectors are called the parametricequations of the a plane.

15. Systems of Linear Equations and Matrices

Alinear equation in nvariables is an equation in the following form:

a1x1 ` a2x2 ` a3x3 ` . . . ` anxn bwherea1 . . . an are thecoefficientsand b is theconstant term.

Thecoefficientsand the constant termmust be constants.

Asolutionof a linear equation is a vector whose components satisfy the equation (subsituting eachxi with the corresponding component).

Asystem of linear equations is a finite set of linear equations with the same variables. A

solutionof the system is a vector that is a solution of all linear equations in the system. Thesolution set is the set of all solutions for the system. Finding the solution set is called solvingthe system.

A system of linear equations with real coefficients has either:

A unique solution (consistent system) Infinitely many solutions (consistent system) No solutions (inconsistent system)

Two linear systems are equivalentif their solution sets are the same.

Linear systems are generally solved by utilizing Gaussian elimination.Take the linear system:

ax ` by mcx ` dy n

Solving this using Gaussian elimination involves creating an augmented matrix:a b mc d n

and putting it in row-echelon form. Using back substitution from there, all the variables can be

solved for, and the solution vector

xy

can be found.

A matrix is in row echelon form if it follows these guidelines:

(1) Rows consisting of zeros only are at the bottom(2) The first nonzero element in each nonzero row (called the leading entry) is in a column to

the left of all other leading entries.

This is an example of a matrix in row echelon form:21


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0 0 0

fifl

Elementary row operations are used to put matrices into row echelon form. The following areelementary row operations:

Interchange two rows

Multiply a row by a constant (nonzero)

Add a row (or any multiple of it) to another

The process of putting a matrix in row echelon form is called row reduction.Matrices are considered row equivalent if a series of elementary row operations can convert one

matrix into the other, and, therefore, if they share a row echelon form, since you can apply theopposite row operations from row echelon form to get the original matrix.

Therankof a matrix is the number of nonzero rows it has in row echelon form.

The Rank Theorem:

Let Abe the coefficient matrix of a system of linear equations in nvariables. If it is a consistentsystem (that is, it has at least one solution), then:

number of free variables n rankpAqor, for every variable, there needs to be another equation to find a single solution.

To simply back substitution, reduced row echelon form can be used.

A matrix is in reduced row echelon form if:

(1) It is in row echelon form(2) The leading entry in each nonzero row is a 1 (called a leading 1)(3) Each column containing a leading 1 has zeroes everywhere else

Gauss-Jordan elimination is similar to Gaussian elimination, except instead of stopping at row

echelon form, it proceeds to reduced row echelon form.

Homogeneous systems of linear equations are systems where the constant term is zero in eachequation. Its augmented matrix is in the form

A 0

.

Homogeneous systems have at least one solution (the trivial solution, which is0

).

If a homogeneous system has less equations than it does variables, it must have infinitely manysolutions.

16. Spanning Sets

Theorem:

A system of linear equations with augmented matrix

A b

is consistent if and only ifb is a linearcombination of the columns in A .

Thespanof a set of vectorstv1, v2, v3, . . . , vku is the set of all linear combinations of that set.If the span is equal to Rn, the set is referred to as a spanning set for Rn.

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17. Linear Independence

A set of vectorstv1, v2, v3, . . . , vku islinearly dependentif scalars exist such that:c1v1 ` c2v2 ` . . . ` ckvk 0

and at least one of the scalars is not 0.Additionally, a set of vectors is linearly dependent if and only if one of the vectors can be expressed

as a linear combination of the others.If a set of vectors is not linearly dependent, they are said to be linearly independent.

Theorem:

If there is a set ofmrow vectors in a matrix A, the set of row vectors is linearly dependent if andonly if rankpAq m

Therefore, any set ofmvectors in Rn is linearly dependent ifm n.

18. Matrices and Matrix Algebra

Amatrixis a rectangular array of numbers. These numbers are called entries or elements.This is an example of a matrix:

a b cd e fg h i

fifl

Thesizeof a matrix is represented as the number of rows (m)the number of columns (n).The above matrix has a size of 3 3.

A 1 mmatrix is called a row matrix(and is also a row vector).A n

1 matrix is called a column matrix (and is also a column vector).

Thediagonal entries of a matrix are those where the column and row it is in are the same.Examples are a11, a22, a33, . . . , ann.

If the matrix has the same number of rows as it does columns, it is a square matrix.A square matrix with zero non-diagonal entries is a diagonal matrix.

A diagonal matrix with all diagonal entries the same is a scalar matrix.Lastly, if the scalar on the diagonal is 1, it is an identity matrix.

Two matrices areequalif their size and corresponding entries are the same.

Adding matrices is as simple as adding each corresponding entry to each other.Scalar multiplication is just as easy: multiply every entry in the matrix by the scalar.

Multiplying two matrices is more complex. If the matrix C AB, where A has size m nand Bhas size n r, then the size ofC is m r.

Each element cij in C is equal to Ai bj.Another way to write this is:

cijn

k1aikbkj

Matrices can be divided into submatrices by partitioning it into blocks.23


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Take, for example, the following matrix:

1 0 0 2 10 1 0 1 30 0 1 4 00 0 0 1 70 0 0 7 2

fiffiffiffiffifl

It can be partitioned as:I BO C

whereI is the 3 3 identity matrix, B is a 3 2 matrix, O is the 2 3 zero matrix, and C is a

2 2 matrix.

Just like with scalar numbers, a matrix Ak is equal to the matrix A multiplied by itselfk times.

IfA is a square matrix, and r and s are non-negative integers, then the following is true:

(1) ArAs Ar`s

(2)pAr

qs

Ars

Thetranspose of a matrix AT is obtained by interchanging the rows and columns of a matrix.Therefore,pATqij Aji.

A matrix is symmetric if its transpose is equal to itself.

Algebraic Properties of Matrix Addition and Scalar Multiplication:

IfA, B, and Care matrices of the same size and c and dare scalars, then the following is true:

A ` B B ` A (Commutativity)

pA

`B

q `C

A

` pB

`C

q(Associativity)

A ` O A A ` pAq O cpA ` Bq cA ` cB (Distributivity) pc ` dqA cA ` dA(Distributivity) cpdAq pcdqA 1A A

Matrices can form linear combinations in the same way vectors do.Similarly, the concept oflinear independence also applies.

Lastly, just as we define the span of a set of vectors,thespan of a set or matrices is the set of all linear combinations of the matrices.

Properties of Matrix Multiplication:

IfA, B, and Care matrices of the appropriatesize such that the operations can be performed,and k is a scalar, then the following is true:

ApBCq pABqC(Associativity) ApB ` Cq AB ` BC(Left

Distributivity)

pA ` BqC AC` BC (RightDistributivity)

kpABq pkAqB ApkBq ImA A AIn (ifA is m n)

Transpose Properties:

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IfA and B are matrices of the appropriate sizeso that the operations can be performed, and kis a scalar, then the following is true:

pATqT A

pA ` BqT AT ` BT pkAqT kpATq pABqT BTAT pArqT pATqr for all non-negative

integers r.

IfA is a square matrix, then A ` AT is symmetric.IfA is a matrix, then AAT and ATAare symmetric for any matrix A.

19. The Inverse of a Matrix

IfA is a square, n nmatrix, it has a unique inverse A1 that is also n nand where thefollowing is true:

AA1 I A1AIf this matrix A

1

exists,Ais invertible.

IfA is a matrix of the form

a bc d

, then ifad bc 0, Ais invertible and the following is true:

A1 1ad bc

d bc a

IfA is an invertible matrix, then A1 is invertible andpA1q1 A. IfA is invertible and c is a nonzero scalar, then cA is invertible andpcAq1 1

cA1.

IfA and B are invertible and have the same size, then AB is invertible andpABq1 B1A1. IfA is invertible, then so is AT, andpATq1 pA1qT. IfA is invertible, then An is invertible for all non-negative integers n, andpAnq1 pA1qn.

IfA is invertible, we can define An aspA1qn pAnq1. Therefore, all properties of matrix powershold for a negative power, provided that the matrix is invertible.

Elementary matricesare those that can be obtained by performing a single elementary rowoperation on an identity matrix.

Performing a row operation can then be expressed by left multiplying an elementary matrix to theoriginal matrix, provided it is of correct size.

All elementary matrices are invertible, and its inverse is also an elementary matrix, with the sametype of row operation performed.

IfA is a square matrix, and a series of elementary row operations can reduce it to I, the same seriesof row operations changesI into A1.

Using this, we can compute the inverse using Gaussian elimination. This is done by reducingA I

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IfAB Ior BA I, A is invertible and B A1

20. LU Factorization

Just as we can factor natural numbers, such as 10

2

5, we can factor matrices as a product ofother matrices. This is called a matrix factorization.

IfAis a square matrix that can be reduced to row echelon form without interchanging any rows,then the Ahas an LU factorization.

An LU factorization is a matrix factorization of the form:

A LUwhereLis a unit lower triangular matrix, and Uis upper triangular.

IfUis the upper triangular matrix obtained when Ais put into row echelon form, then Lis the

product of the elementary matrices representing each row operation.Lcan also be obtained by using the multipliers of each row operation of the form Ri kRj , with

each multiplier being thepi, jq entry in the Lmatrix.The row operations must be done in top to bottom, left to right order for this process to apply.

A matrix P that arises from interchanging two rows in the identity matrix is called apermutationmatrix.

The transpose of a permutation matrix is its inverse.

IfA is a square matrix, then it has a factorization A PTLU, where Pis a permutation matrix,and U and Lare as defined above.

Every square matrix has a PTLU factorization.

21. Subspaces, Dimensions and Basis

Asubspaceis a collection of vectors inside the space Rn, so that it contains the zero vector 0, it isclosed under addition, and it isclosed under scalar multiplication.

Therefore, for any set of vectors in a subspace, all linear combinations of those vectors are in thesame subspace.

For this reason, the span of a set of vectors forms a subspace in Rn itself.The subspace formed by the span of a set of vectors is the subspace spanned by that set.

Therow space is the subspace spanned by the rows of a matrix, and the column space, by thecolumns.

Matrices that are row equivalent have equivalent row spaces.Abasisfor a subspace is a set of vectors that spans it, and is linearly independent.

Thestandard basis is a basis that consists of the standard unit vectors. An example would be:

e1, e2, . . . , en for the space Rn.

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The solution to an equation of the form Ax 0is referred to as the null spaceof the matrix A,and is a subspace ofRn, where n is the number of columns in the matrix.

For any system of linear equations of the form Ax b, it either has no solutions, one uniquesolution, or infinitely many solutions.

Two bases for a subspace must have the same number of vectors.

This leads to the definition of dimension, which is the number of vectors in the basis for asubspace.

The row and column spaces of a matrix must have the same dimension.

We can now redefinerankas the dimension of a matrixs row/column space.A matrixs transpose has the same rank as itself.

Thenullity of a matrix is defined as the dimension of its null space.rankpAq ` nullitypAq n, where n is the number of columns.

Using the notion of basis, one can denote any vector in a subspace as a linear combination of itsbasis vectors. The coefficients of the linear combination form thecoordinates of the vector with

respect to the basis.

22. Linear Transformations

A transformation T : Rn Rm is a linear transformation ifTpu ` vq Tpuq ` Tpvq

andTpcvq cTpvq.

Amatrix transformation of the form Ax

TApxq

is a linear transformation as well.All linear transformations can be expressed as a matrix transformation where each column of the

standard matrix is the transformation of the standard unit vectors.

Composite transformationsarise when apply a transformation to another transformation. Theirstandard matrices are related in that the matrix of the composite is the product of the matrix.

Two transformations areinverse to each other if both composites (in either order) yield theidentity matrix.

The matrix of an inverse transformation is the inverse of the original matrix.

23. Determinant of a Matrix

Only square matrices have determinants.

Thedeterminantof a square matrix is:

det A n

j1aijCij

or

det A n

i1aijCij

whereCij is thepi, jqcofactor.Thecofactoris defined asp1qi j det Aij.

Aij is the matrix A with the ith row and jthcolumn omitted.

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The determinant for a 1 1 matrix is the valueof its element.

For a matrix Aand any square matrix B,

detpABq pdet Aqpdet Bq.

A matrix is invertible if and only if itsdeterminant is zero.

For any square matrix A:

det A det AT

and

detpA1q 1det A

(if the matrix is invertible)s

The inverse of any invertible matrix is:

A1

1

det Aadj A

Theadjoint matrix is the transpose of thecofactor matrix.

24. Eigenvalues and Eigenvectors

Aneigenvalue is a scalar (denoted by ) such that:

Ax xxis called an eigenvectorof the matrix A.

All eigenvectors corresponding to an eigenvalue forms an eigenspace.The eigenvalues of a matrix Aare the solutions to:

detpA Iq 0

Thealgebraic multiplicity of an eigenvalue is the number of times it is a root in thecharacteristic equation.

Thegeometric multiplicity is the dimension of its eigenspace.Since the determinant of a triangular matrix is equal to the product of the diagonal entries, the

eigenvalues of a triangular matrix are the values on its diagonal.n is an eigenvalue ofAn for any integer n, with the same corresponding eigenvector, ifis an

eigenvalue ofA.

If a vectorx can be expressed as a linear combination of the eigenvectors ofA, the following is true:

Akx c1kv1 ` c2kv2 ` . . .

The eigenvectors corresponding to unique eigenvectors are linearly independent.

25. Similarity of Matrices

If a matrix Pexists so that P 1AP B, A B.This issimilarityof matrices.

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Similarity is transitive.The following is true of similar matrices:

Their determinants are the same. They are either both invertible or not invertible. Their rank is the same. Their characteristic polynomial, and, therefore, their eigenvalues, are the same.

A matrix is diagonalizable if it is similar to a diagonal matrix.From this, it is diagonalizable if it the same number of unique eigenvectors, and, therefore, distinct

eigenvalues as it does rows/columns.If a matrix is diagonalizable, its diagonal matrix D has entries that are its eigenvalues, and its P

matrixs columns are the corresponding eigenvectors, in order.

26. OrthogonalityA set of vectors is an orthogonal setif every vector is orthogonal (its dot product is zero) to every

other vector in the set.An orthogonal set is linearly independent.

Anorthoganal basis is a basis that is an orthogonal set.

For any vectorx in a subspace defined by the orthogonal basis v1, v2, . . ., it is equal to

c1v1 ` c2v2 ` . . . .where:

ci x vivi vi

Anorthonormal set is an orthogonal set of unit vectors.Anorthogonal basis is defined similarly.

The columns of a matrix Q are an orthonormal set if and only if

QTQ Iand this is called an orthogonal matrix.

For every orthogonal matrix Q: Q1 QT Qx x Qx Qy x y Q1 is orthogonal det Q 1 The absolute value of its eigenvalues is one. Q1Q2 is orthogonal ifQ1 and Q2 are.

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A vector isorthogonal to a subspaceif it is orthogonal to every vector inside of the subspace(which is equivalent to being orthogonal to each of its basis vectors).

The set of all vectors orthogonal to a subspace is its orthogonal component.

For a matrix A:

prowpAqqK nullpAq

andpcolpAqqK nullpATq

These four subspaces are called the fundamental subspaces ofA.

Theorthogonal projection of a vector onto a subspace is the sum of the projections of the vectoronto each of the orthogonal basis vectors.

Thecomponent of the vector orthogonal to the space is the difference between the vectorand its projection.

The Gram-Schmidt Process: This is a process that takes a basis for a subspace and makes anorthogonal one. it is done by taking each basis vector one at a time, and subtracting theprojections of it onto the previous basis vectors (taking the perpendicular component of the vector

to the previous basis vectors).

This leads to the QR factorization.The QRfactorization can be done with any matrix with linearly independent columns.

It is factored A QR, where Q is a matrix with orthonormal columns and R is an invertible uppertriangular matrix.

A matrix isorthogonally diagonalizable if the diagonalizing matrix is orthogonal.

Orthogonally diagonalizable matrices are symmetric, and vice versa.Distinct eigenvalues of a symmetric matrix have orthogonal eigenvectors.

Based on the Spectral Theorem, any real symmetric matrix Acan be written as A QDQT, andsince the diagonal entries ofD are the eigenvalues ofA, the matrix Acan be written as:

A 1q1q1T ` 2q2q2T ` . . .This is called thespectral decomposition ofA.

27. Vector Spaces

Avector space is a set where addition and scalar multiplication are defined in some fashion, andthe following axioms are true for all vectors u, v, andw and for all scalars c and din the set:

(1) It is closed under addition.(2) It is commutative for addition.(3) It is associative for addition.(4) A zero vector exists, and is the additive identity.(5) For each vector uthat exists, there is an opposite,u, such that their sum is0.(6) It is closed under scalar multiplication.(7) It is distributive such that the scalar can be distributed to the sum of vectors.

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(8) It is distributive such that the vector can be distributed to the sum of scalars.(9) cpduq pcdqu

(10) 1u uThe operations addition andscalar multiplicationcan be defined in any way so that the axioms arefulfilled, and the vectors in the set may be anything so that the axioms are fulfilled as well.For any vector space with u and cin it, the following are true as well:

0u

0

c0 0 p1qu u Ifcu 0, either c 0 oru 0.

A subset is called a subspaceof a vector space if it is also a vector space, and has the same scalars,addition, and scalar multiplication definitions.

Following from this, ifV is a vector space, and W is a subset (and is not empty) ofV , W is asubspace ofV if and only ifWis closed under addition and scalar multiplication.

A subspace has the same zero vector as its containing space(s).

The span of a set of vectors in a vector space is the smallest subspace containing those vectors.

Linear combinations, linear independence, and basis is defined in the same way as conventionalvectors.

For a basis of a vector space: any set with more vectors than the basis is linearly dependent, andwith fewer, it cannot span the vector space.

Vector spaces are finite-dimensionalif their basis has a finite amount of vectors; otherwise, theyare infinite-dimensional.

28. Change of Basis

Thechange-of-basis matrix fromBtoC PCB is the matrix whose columns are the coordinatevectors of the original basis vectors into the new basis.

The matrix PCB has the following properties: PCBrxsB rxsC PCB is unique for the above property. PCB is invertible and its inverse is PBC

By expressing the basis vectors for two bases BandCin terms of another base (call theseexpressionsB and C) , row reduction of

C B

yields

I PCB

.

29. Kernel and Range

Thekernelof a linear transformation is the set of all vectors mapped by a linear transformation.Therange is the set of all vectors that are images created by a linear transformation.

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Therankof a linear transformation is the dimension of its range.Thenullityof a linear transformation is the dimension of its kernel.

The rank and nullitys sum is equal to the dimension of the original vector space of thetransformation.

A transformation is called one-to-one if every mapping is distinct. If the range of a lineartransformation is equal to the ending space, it is onto.

A linear transformation is one-to-one if and only if its kernel ist0u.If the dimension of both spaces is the same for a linear transformation, either it is one-to-one and

onto, or neither.A linear transformation is invertible if and only if it is one-to-one and onto.

A linear transformation that is both one-to-one and onto is anisomorphism.

30. Inner Products

Aninner product is an operation on a vector space that assigns a real number to every pair of

vectors so that the following are true for any vector u and v, and for any scalar c:(1)xu, vy xv, uy(2)xu.v ` wy xu, vy ` xu, wy(3)xcu, vy cxu, vy(4)xu, uy 0, and only equals 0 when u 0

Thelength or normof a vector v is v axv, vy

The distance between two vectors is the norm of their difference.Two vectors areorthogonalif their inner product is zero.

31. Norms

Anorm on a vector space associates a single vector with a real number so that the following aretrue:

(1) The norm of a vector is always greater than or equal to zero. The only time it is zero is whenthe vector is the zero vector.

(2) cv |c|v(3) The norm of the sum of two vectors is less than or equal to the sum of the norms of those

two vectors.

The sum norm, also called the 1-normis the sum of the absolute value of the vectors components.The8-normor max norm is the greatest of the absolute values of the components of a vector.

Adistance functionis defined as:

distancepu, vq u vAmatrix normassociates each square matrix A with a real number so that the following is true,

in addition to the vector norm conditions (with matrices instead):

AB AB32


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32. Least Squares Approximation

The best approximation to a vector in a finite-dimensional subspace is its projection upon thesubspace.

IfAis an m nmatrix andb is in Rn, a least squares solutionofAx b in Rn so that:b Ax b Ax

for allxin Rn.

The solution,xis a unique least square solution ifAtAis invertible. If so, the following is true:

x pATAq1ATbThe least squares solution of a QR factorized matrix is:

x R1QTb

Thepseudoinverseof a matrix Awith linearly independent columns is:

A` pATAq1ATThe pseudoinverse has the following properties:

AAÀ A AÀA` A` AA` and AÀare symmetric.

33. Singular Value Decomposition

Thesingular values of a matrix Aare the square roots of the eigenvalues ofATA. They aredenoted by 1 . . . n.

Thesingular value decomposition is a way of factoring a m nmatrix Ainto the formA UVT whereU is an m morthogonal matrix, V is an n northogonal matrix, and is an

m nmatrix of the form:

D OO O

whereD is a diagonal matrix whose entries are the nonzero singular values ofA.V is constructed from the eigenvectors ofATAso that each column is the corresponding eigenvector

to the singular value.U is constructed from the eigenvectors ofAAT so that each column is the corresponding eigenvector

to the singular value.

The matrix Acan then be written as:

1u1v1T ` 2u2v2T ` . . .

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34. Fundamental Theorem of Invertible Matrices

Each of the following statements implies the others:

Ais invertible, where Ais a n nmatrix.

Ax

b has a unique solution for every b.

Ax 0has only the trivial solution. The reduced row echelon form ofA is I. Ais a product of elementary matrices. rankpAq n. nullitypAq 0. The column vectors ofAare linearly independent. The column vectors ofAspan Rn. The column vectors ofAform a basis for Rn. The row vectors ofAare linearly independent. The row vectors ofAspan Rn.

The row vectors ofAform a basis for Rn.

det A 0. 0 is not an eigenvalue ofA. T is invertible where T :V Wis a linear transformation from basis Bto basis C. T is one-to-one. T is onto. kerpTq t0u rangepTq W 0 is not a singular value ofA.

mcla concise review

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