Vector Spaces - PowerPoint Presentation

Slide1

Vector Spaces & Subspaces

Kristi SchmitSlide2

Definitions

A subset W of vector space V is called a

subspace

of V

iff

The

zero vector of V is in W.

W

is closed under vector addition, for each

u

and

v

in W, the sum

u

+

v

is in W.

W

is closed under multiplication by scalars, for each

u

in W and each scalar

c

, the vector

c

u

is in W.

Any subspace W of vector space V is a vector space. Slide3

Example 1

Let W be the set of all vectors of the form shown, where

a

and

b represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space.Slide4

Response

The zero vector of V is not in W because of the -1 in the subset.

Therefore the subset fails the first property of a subspace.

Thus, W is not a subspace of V and therefore is not a vector space.Slide5

Definitions

If

v

1

,….,

vp are in an n-dimensional vector space over the real numbers, R

n, then the set of all linear combinations of v1 ,…., vp is denoted by Span{v1

,….,

v

p } and is called the subset of Rn spanned by v1 ,…., vp. That is, Span{v1 ,…., vp } is the collection of all vectors that can be written in the form c1v1+ c2v2+ …+ cpvpwith c

1,…, cp scalars.Slide6

Example 2

Let W be the set of all vectors of the form shown, where

a

and

b

represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space. Slide7

Response

the

set of all linear combinations of

v

1

,….,

v

p