LCMPLETej^—\

Math 261 Lecture Notes: Sections 1.3 and 1.4 w ^/] "7-\Instructor: A.E.C'arv > y

Definition 1. Vectors in Rn. Euclidean n-space (or simply n-space) i.s denoted Rn and is defined to

be the set of n-dimensional vectors given by/

Zl

= <

#2

. .•''•'".

xi e E, for i = 1,2,... ,n

~VAa5 13 a/\ ^*l -^li/AJVS\ */w^\ f^aSfi/Example 1. Give examples of vectors in Euclidean 2-space and Euclidean 3-space.

u =

- ll f—•n *•

u

^JTT

Definition 3. The scalar multiple of a vector v is given by cv, where c G R. It represents a vector

that is scaled by that factor and pointing in the same direction as the vector v.

Properties of Vectors. Let u,v, and w be vectors in R" and let c and d be scalars. The following-

properties hold:

(1) u + v = v + u

(2) (u + v) + w = u + (v -I- w)(3) The zero vector, 0 (aka the additive identity), exists and satisfies 0 + u = u.

(4) Every vector u has an additive inverse, denoted by —u, such that u + (—u) = 0.(5) c(u + v) = cu + cv

(6) (c + d)u = cu + du

(7) c(du) = (cd)u(8) (l)u = u

_^•a

//

I

(a -•"V

(AZ

M '

/

Math 261 Lecture Notes

•^

-i

"~" r-1•* -^ ^ , .

(/> TM ~"7

t&_

.- H1^ - \ 3

l_ J U — L JSections 1.3-1.1

Example 2. When adding two vectors, it is helpful to think of one vector becoming an extension of the

other. We add vectors "head-to-tail," as shown in Figure 1.

—l -3 -2 -1

V

-3

Example 3. Let u = . and v =

v\j - 3 cC -i\f

? 3

#»* —1

1 i 1*

2 ' 2- C*

1A L^J

3

3 4

Figure 2

1

3

2

•v

-i 41>>L i 7-7~V^ / x

—1 -3 -2 -i "> j 2 } 4

-i CT *>

-3

—1

. Compute w = 3u —kv.

We call the vector w a linear combination of u and v.

Instructor: A.E.Carv Page 2 of 9

Math 2G1 Lecture Notes Sections 1.3-1.4

Definition 4. A vector written as a combination of other vectors using addition and scalar multiplication is called a linear combination. More formally, for the set of vectors S = {vi, V2,.... vj<} inR" and scalars c\, c-y,..., c/. € R. an expression of the form

k

C1V1 +c2v2 H hc/,vk = 22civi

is a linear combination of the vectors in S. Any vector that can be written in this form is called

a linear combination of S.

Definition 5. The set of all linear combinations of a set of vectors S —{vi, Vg, •••, vn} is called thespan of that set. We write Span(5) or Spanjvi, V2,. •., vn}.

Linear Independence of

{ red, yellow, blue }

Instructor: A.E.Gary

Linear Combinations of

{red, yellow, blue}Span(red, yellow, blue)

Page 3 of 9

Math 261 Lecture Notes Sections 1.3-1.4

Example 4. Let's consider the span of some familiar vectors...

• What is the span of two vectors in R2? Does it matter if they are parallel?

^s. \\/\-«- ^• What is the span of two vectors in R ? Does it matter if they are parallel?

« GOWN '\<S v^-«L. pW>«ft£^ OV^^' "^ vWoV~. ^ VCC.lV/5?

• What is the span of three vectors in R3? Does it matter if they lie in the same plane?

J.-f "Wj^nA O© ^ot ^v^^^ANa^ <S^«a\-^ nlo'/L4; ~H\£L\-/" 5Q<Sc\A '\ 5

2 r,v 1B>3?• What is the span of a single vector in R~ or

Check out this Intel webpage on linear combinations and color models:

http://software.intel.com/sites/products/documentation/hpc/ipp/ippi/ippi_ch6/ch6_color_

models.html

Instructor: A.E.Cary Page 4 of 9

Math 261 Lecture Notes

Definition 6. A vector equation is of the form

3,'iai + a;2a2 H + .T„an = b

where X\,xn, • • •, %n are constants and ai,a2,..., an, and b are vectors.

This eciuation has the same set of solutions as the augmented matrix

c o\ a w. w

\z£cYb^ai^a2 an b

Sections 1.3-1.4

The vector b can be generated as a linear combination of ai, a2,..., a„ if and only if there exists a

solution to the linear system corresponding to Matrix 6.

Example 5. Use the definition above to find the solutions to the vector eciuation

* K2. \ A

7. C h

3 IO 1

Y 2 2

2 + X2 6 = 3

_3_ 10 4

Xy 2

*,-3, %.

Circle;

2-

+K) o

i

Instructor: A.E.Cary

j^-^p-2-

0*

-2R2 + M —a*R\

-1 HZ. v P-3 -^ (2.3

Page 5 of 9

Math 261 Lecture Notes Sections 1.3-1.-1

Example 6. Let ai =

1 "-2] h

0 , a2 = 1 and b = -3

_2 7 -5

. For what value(s) of h is the vector b in

Span{a!.a2}?

(An equivalent question would be to ask what value(s) of h make the vector b in the plane generated byai and a2).

a 0

c

G K

t^c X, o.w\ 9^ ^3+ 6aoVk ^C^T -X,a, + %^C\^^ y

X, %.

&£l +a3^>^.3

V

0

O O \4v2V

Instructor: A.E.Gary

TWa.s 1a - "Z- ,

~XVjuttxW^-5

</r£ Wm

-2-

-5

Page 6 of" 9