Department

Mathematics

Course Code

MATA23H3

Professor

Sophie Chrysostomou

Subspaces, Basisand Rank ofMatrices

DEFINITION: Let W⊂Rn.Wis calledaasubspaceof Rnif:

i) Wis nonempty,

ii) if u,v∈W,then u+v∈W,(closureunder vector addition)

iii) if u∈Wand r∈R,then ru∈W.(closure under scalar multiplication)

EXAMPLE:If Ais anm×nmatrix and Nis the nullspace ofA,then Nis asubspace of

Rn.

EXAMPLE:If W=sp(v1,v2,· · · ,vm)wherev1,v2,· · · vm∈Rn,then Wis asubspace

of Rn.

www.notesolution.com

EXAMPLE:If W={[x1,x2,x3,x4]∈R4

x1=x3−x4,x2=x3+x4},determine if Wis

asubspace ofR4.

EXAMPLE:Is W={[x1,x2,x3]∈R3

x1+x3=x2+3}asubspace ofR3?

2c

2011 bySophie Chrysostomou

www.notesolution.com

DEFINITION: Let Wbeasubspace ofRn.If B={b1,b2,· · · ,bk}is asubset ofW,

then wesaythat Bis abasis for Wif everyvectorin Wcan bewrittenuniquelyas a

linear combinationofthe vectorsin B

The pluralforthe word basis is ”bases”.

EXAMPLE:Let W=

x

y

x

−y

x, y∈R

Wis asubspace of R4.

Is B=

1

1

1

−1

,

1

0

1

0

,

0

−1

0

1

abasis forW?

3c

2011 bySophie Chrysostomou

www.notesolution.com

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