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Chapter 1: Problem 2
Let \(u=(2,-7,1), v=(-3,0,4), w=(0,5,-8) .\) Find: (a) \(3 u-4 v\) (b) \(2 u+3 v-5 w\)
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The set of solutions to the system of linear equations $$ \begin{array}{r} x_{1}-2 x_{2}+x_{3}=0 \\ 2 x_{1}-3 x_{2}+x_{3}=0 \end{array} $$ is a subspace of \(R^{3}\). Find a basis for this subspace.
Exercises \(29-34\) require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3. (a) Prove that if \(W_{1}\) and \(W_{2}\) are finite-dimensional subspaces of a vector space \(V\), then the subspace \(W_{1}+W_{2}\) is finite-dimensional, and $\operatorname{dim}\left(\mathrm{W}_{1}+\mathrm{W}_{2}\right)=\operatorname{dim}\left(\mathrm{W}_{1}\right)+\operatorname{dim}\left(\mathrm{W}_{2}\right)-\operatorname{dim}\left(\mathrm{W}_{1} \cap \mathrm{W}_{2}\right)\(. Hint: Start with a basis \)\left\\{u_{1}, u_{2}, \ldots, u_{k}\right\\}\( for \)\mathrm{W}_{1} \cap \mathrm{W}_{2}$ and extend this set to a basis $\left\\{u_{1}, u_{2}, \ldots, u_{k}, v_{1}, v_{2}, \ldots, v_{m}\right\\}\( for \)\mathrm{W}_{1}\( and to a basis \)\left\\{u_{1}, u_{2}, \ldots, u_{k}, w_{1}, w_{2}, \ldots, w_{p}\right\\}$ for \(\mathrm{W}_{2}\). (b) Let \(W_{1}\) and \(W_{2}\) be finite-dimensional subspaces of a vector space \(\mathrm{V}\), and let \(\mathrm{V}=\mathrm{W}_{1}+\mathrm{W}_{2}\). Deduce that \(\mathrm{V}\) is the direct sum of \(\mathrm{W}_{1}\) and \(W_{2}\) if and only if \(\operatorname{dim}(V)=\operatorname{dim}\left(W_{1}\right)+\operatorname{dim}\left(W_{2}\right)\).
Let \(M\) be a square upper triangular matrix (as defined on page 19 of Section 1.3) with nonzero diagonal entries. Prove that the columns of \(M\) are linearly independent.
Prove: For any complex numbers \(z, w \in \mathbf{C},|z w|=|z||w|\)
Consider a moving body \(B\) whose position at time \(t\) is given by \(R(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}+3 t \mathbf{k} .\) [Then \(V(t)=d R(t) / d t \text { and } A(t)=d V(t) / d t \text { denote, respectively, the velocity and acceleration of } B .]\) When \(t=1,\) find for the body \(B:\) (a) position; (b) velocity \(v\) (c) speed \(s\) (d) acceleration \(a\)
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