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Lecture 5

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Department
Mathematics
Course
MATA23H3
Professor
Kathleen Smith
Semester
Winter

Description
Definition: The linear system Ax = 0 is said to be homoge- neous. The solution x = 0 is called the trivial solution. Nonzero solutions are called nontrivial solutions. Theorem: If 1 ,v2, ··· ,vkare solutions of the homogeneous linear Xk system Ax = 0, the linear combination λiv is also a solution i=1 for any scalars 1 ,λ2, ··· ,λk. Definition: A nonempty subset W of R is called a subspace of R if the following conditions hold. (i) For all u, w ∈ W, u + w ∈ W. (closed under vector addition) (ii) For all w ∈ W and λ ∈ R, λw ∈ W. (closed under scalar multiplication) Corollary: Let W be the solution set of a homogeneous linear sys- n n tem Ax = 0, where x ∈ R . Then W is a subspace of R . Definition: The subspace, W = { 0 } of R is called the zero subspace. n n Definition: All subspaces of R , other than R itself, are called n proper subspaces of R . Theorem: If W = sp(w ,w , 1·· 2w ) is thekspan of k vectors in R , then W is a subspace of R . n Definition: We say that the vectors w ,w ,1·· 2w , as ik the the- orem, generate or span the subspace W = sp(w ,w , ··· ,w ). 1 2 k    a a ··· a   11 12 1k    21 a22 ··· a2k Let A =  ∈ M (R).   n,k  .      an1 an2 ··· ank The span of the row vectors of A, sp([a ,a , ··· ,a ], ··· ,[a ,a ,··· ,a ]), 11 12 1k n1 n2 nk is called the row space of A. It is a subspace of R . The span of theolum vector of   a11  a1k          a21  a2k sp  ···  ,  .   .   .   .          an1 ank is called the column space of A. It is a subspace of R . The solution set of Ax = 0 is called the nullspace of A. It is a subspace of R . Remark: A linear system Ax = b has a solution if and only if b is in the column space of A.
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