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Lecture

Subspace, basis, rank

12 Pages
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Department
Mathematics
Course Code
MATA23H3
Professor
Sophie Chrysostomou

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Subspaces, Basisand Rank ofMatrices
DEFINITION: Let WRn.Wis calledaasubspaceof Rnif:
i) Wis nonempty,
ii) if u,vW,then u+vW,(closureunder vector addition)
iii) if uWand rR,then ruW.(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,· · · vmRn,then Wis asubspace
of Rn.
www.notesolution.com
EXAMPLE:If W={[x1,x2,x3,x4]R4
x1=x3x4,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, yR
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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Description
Subspaces, Basis and Rank of Matrices n n DEFINITION: Let W R . W is called a a subspace of R if: i) W is nonempty, ii) if u,v W, then u + v W, (closure under vector addition) iii) if u W and r R, then ru W. (closure under scalar multiplication) EXAMPLE: If A is an mn matrix and N is the nullspace of A, then N is a subspace of R . EXAMPLE: If W = sp(v ,v ,1 2v ) wheme v ,v ,1v 2 m R , then W is a subspace n of R . www.notesolution.com EXAMPLE: If W = {[x ,x ,x ,x ] R x = x x , x = x + x }, determine if W is 1 2 3 4 1 3 4 2 3 4 4 a subspace of R . 3 3 EXAMPLE: Is W = {[x ,x ,x ] R 1 + 2 = 3 + 3} a ub1pace 3f R ?2 2 2011 by Sophie Chrysostomou www.notesolution.com
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