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

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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 λ u ∈ Nul A . A ( λ u ) = λA u = λ = . 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 and...

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