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Question: 7 please...

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Ve will now define four important subspaces associated with any matrix: efinitions/Theorem-The Four Fundamental Matrix Spaces: t A be an m x n matrix. The rowspace of A is the Span of the rows of A. The columnspace of A is the Span of the columns of 4. The nullspace of A is the set of all solutions to At - 0m rowspace(A) Span(1, 2,...,Im), 7: Prove that nullspace(A) ( ER Az Om) is a subspace of Rn by definition. Proof 1. Consider On, IR, then we have Ai- AO On; hence, On, E nullspace(A). 2. For all u, E nullspace(A), Au On, Av Om. Then A(u+) Au+ Au-Om + 3. For all ขึ้ E nullspace(A) and hence, Au -0,ni for all r E R, we have A(r,u) Thus, we conclude that ti+fe nullspace(A). r(A0 Thus, we have r E nullspace(A). 7 please
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