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# Subspace, basis, rank

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University of Toronto Scarborough

Mathematics

MATA23H3

Sophie Chrysostomou

Winter

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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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