Linear Algebra

Santanu Dey

January 4, 2016

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Cartesian coordinate space

René Descarte (1596-1650)

French philosopher, mathematician, physicist, and writer.

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n-dimensional Cartesian coordinate space Rn

∼= R× . . .× R (n factors)Rn is the totality of all ordered n-tuples (x1, . . . , xn) wherexi ∈ Rfor n = 2 is the (x , y) ∈ R2

πi : Rn → R defined by

πi((x1, . . . , xn)) = xi

is called the i th coordinate function or i th coordinateprojectionGiven a function f : A→ Rn,define fi := πi ◦ fthese fi : A→ R completely determines f

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Algebraic structure of Rn

For x = (x1, . . . , xn),y = (y1, . . . , yn) define

x + y = (x1 + y1, . . . , xn + yn)

Note: usual laws of addition,

0 = (0, . . . ,0),−x = (−x1, . . . ,−xn)

Scalar multiplication: αx := (αx1, . . . , αxn)

1. associative: α(βx) = (αβ)x2. distributive: α(x + y) = αx + αy3. identity: 1x = x

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Linear maps

Definitionf : Rn → Rm is said to be a linear map iff (αx + βy) = αf (x) + βf (y).

Examples: Projection map πi , multiplication by scalardot product by a fixed vector; what about the converse?f : Rn → Rm is linear iff fi s are linearDistance travelled is a linear function of time when velocityis constant. So is the voltage as a function of resistancewhen the current is constant. The logarithm of the changein concentration in any first order chemical reaction is alinear function of time.|x |, xn (n > 1), sin x , etc. are not linear

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Exercise:

(i) Show that if f is a linear map then

f (k∑

i=1

αixi) =k∑

i=1

αi f (xi).

(ii) Show that the projection on a line L passing through theorigin defines a linear map of R2 to R2 and its image isequal to L.

(iii) Show that rotation through a fixed angle θ is a linear mapfrom R2 → R2.

(iv) By a rigid motion of Rn we mean a map f : Rn → Rn suchthat

d(f (x), f (y)) = d(x,y),

where d(x,y) =√∑n

i=1(xi − yi)2, x,y ∈ Rn

Show that a rigid motion of R3 which fixes the origin is alinear map.

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Structure of linear maps

L(n,m) = Set of all linear maps from Rn to Rm

For f ,g ∈ L(n,m) define αf and f + g by

(αf )(x) = αf (x); (f + g)(x) = f (x) + g(x)

If f ∈ L(n,m) and g ∈ L(m, l), then g ◦ f ∈ L(n, l)If f ,g ∈ L(n,1), then define fg : Rn → R by

(fg)(x) = f (x)g(x).

Does fg ∈ L(n,1)?Let ei = (0, . . . ,0,1,0, . . . ,0) (standard basis elements). Ifx ∈ Rn, then x =

∑ni=1 xiei .

If f ∈ L(n,m), then

f (x) =∑

i

xi f (ei)

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if given v1, . . . , vn ∈ Rm, define a (unique) linear map f byassigning f (ei) = vi

Examples:1. Given a f ∈ L(n,1), if we put u = (f (e1), . . . , f (en)), then

f (x) =∑

i xi f (ei) = u.x2.

a11x1 + a12x2 + . . . + a1nxn = b1

a21x1 + a22x2 + . . . + a2nxn = b2

. . . . . .

am1x1 + am2x2 + . . . + amnxn = bm

Set of all solutions of j th equation is a hyper plane Pj in Rn.Solving the system means finding

P1 ∩ . . . ∩ Pm.

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On the other hand the lhs of each of these equations can bethought of as a linear map Ti : Rn → R. Together, they defineone function

T ∈ L(Rn,Rm)

such that T = (T1, . . . ,Tm).

Determining x ∈ Rn such that T (x) = b,where b = (b1, . . . ,bm)

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Matrix representation

(x1, x2, . . . , xn) is a row vectorx1x2...

xn

= (x1, x2, . . . , xn)T ( where T stands for

transpose) is called a column vector and elements of Rn

are denoted by this.Given linear map f : Rn → Rm we get n column vectors (ofsize m) viz., f (e1), . . . , f (en). Place them side by side:For instance, if f (ej) = (f1j , f2j , . . . , fmj)

T , then we obtain

Mf =

f11 f12 . . . f1nf21 f22 . . . f2n...

...fm1 fm2 . . . fmn

This array is called a matrix with m rows and n columns.

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We say matrixMf is of size m × n.Notation: Mf = ((fij))Matrices are equal if their sizes are the same and theentries are the same.If m = 1 we get row matrices; if n = 1 we get columnmatrices

M : L(n,m)→ Mm,nf 7→ Mf

is one-one and called matrix representation of linear maps

Mf+g =Mf +Mg ; Mαf = αMf

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Examples:

1. MId = I or In where

I =

1 0 . . . 00 1 . . . 0...

...0 0 . . . 1

= ((δij))

δij = 1 if i = j and = 0 otherwise(Kronecker delta)

2. Linear T : R2 → R2 which interchange coordinates is

represented by(

0 11 0

)3. Corresponding to multiplication by α ∈ R is the diagonal

matrix D(α, . . . , α) = ((αδij))

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