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Chapter 5: Q32E (page 233)
(a) Consider an matrix A such that . It is necessarily true that? Explain.
(b) Consider an matrix A such that . Is it necessarily true that ? Explain.
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Let Abe anmatrix. Is the formulanecessarily true? Explain.
By using paper and pencil, find the least squares of the system , where and. Verify that the vector is perpendicular to the image of A.
Consider the subspace of . Where . Find a basis of , and draw a sketch illustrating the formulain this case.
Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.
8.
Question: In Exercises 1 and 2, you may assume that\(\left\{ {{{\bf{u}}_{\bf{1}}},...,{{\bf{u}}_{\bf{4}}}} \right\}\)is an orthogonal basis for\({\mathbb{R}^{\bf{4}}}\).
1.\({{\bf{u}}_{\bf{1}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\\{ - {\bf{1}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{2}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{5}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{3}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{4}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{5}}\\{ - {\bf{3}}}\\{ - {\bf{1}}}\\{\bf{1}}\end{aligned}} \right]\),\({\bf{x}} = \left[ {\begin{aligned}{*{20}{c}}{{\bf{10}}}\\{ - {\bf{8}}}\\{\bf{2}}\\{\bf{0}}\end{aligned}} \right]\)
Write x as the sum of two vectors, one in\({\bf{Span}}\left\{ {{{\bf{u}}_1},{{\bf{u}}_2},{{\bf{u}}_3}} \right\}\)and the other in\({\bf{Span}}\left\{ {{{\bf{u}}_{\bf{4}}}} \right\}\).
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