l10 - Math 20F Linear Algebra Lecture 10: 4.2 Null space...

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Unformatted text preview: Math 20F Linear Algebra Lecture 10: 4.2 Null space and Column space.. The null space of an m n matrix A is the set of all solutions of the homogeneous equation A x = ; Nul A = { x R n ; A x = } . Theorem . The null space of an m n matrix A is a subspace of R n . Proof . We must verify the three properties (a), (b), (c) in the definition of subspace. (a) Nul A since A = . (b) If u , v Nul A , show that u + v Nul A . A ( u + v ) = A u + A v = + = . (c) If u Nul A , show that u Nul A . A ( u ) = A u = = . Example 1 . Find an explicit description of Nul A where A = 3 6 6 3 9 6 12 13 0 3 . Sol Row reduction to solve A x = 0; 3 6 6 3 9 0 6 12 13 0 3 0 (1) / 3 1 2 2 1 3 0 6 12 13 0 3 0 (2)- 6(1) 1 2 2 1 3 0 0 1- 6- 15 0 (1)- 2(2) 1 2 0 13 33 0 0 1- 6- 15 0 Hence A x = x 1 + 2 x 2 + 13 x 4 + 33 x 5 = 0 x 3- 6 x 4- 15 x 5 = 0 . x 2 ,x 4 ,x 5 are free so the sol. is x 1 x 2 x 3 x 4 x 5 = - 2 x 2- 13...
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l10 - Math 20F Linear Algebra Lecture 10: 4.2 Null space...

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