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Chapter 1: Problem 7
Find \(u \cdot v\) where: (a) \(u=(2,-5,6)\) and \(v=(8,2,-3)\) (b) \(u=(4,2,-3,5,-1)\) and \(v=(2,6,-1,-4,8)\)
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Let \(W\) be a subspace of a (not necessarily finite-dimensional) vector space \(V\). Prove that any basis for \(W\) is a subset of a basis for \(V\).
Recall from Example 3 in Section \(1.3\) that the set of diagonal matrices in \(\mathrm{M}_{2 \times 2}(F)\) is a subspace. Find a linearly independent set that generates this subspace.
Let \(u, v\), and \(w\) be distinct vectors of a vector space \(\mathrm{V}\). Show that if \(\\{u, v, w\\}\) is a basis for \(\mathrm{V}\), then $\\{u+v+w, v+w, w\\}\( is also a basis for \)\mathrm{V}$.
Determine which of the following sets are bases for \(\mathrm{P}_{2}(R)\). (a) \(\left\\{-1-x+2 x^{2}, 2+x-2 x^{2}, 1-2 x+4 x^{2}\right\\}\) (b) \(\left\\{1+2 x+x^{2}, 3+x^{2}, x+x^{2}\right\\}\) (c) \(\left\\{1-2 x-2 x^{2},-2+3 x-x^{2}, 1-x+6 x^{2}\right\\}\) (d) \(\left\\{-1+2 x+4 x^{2}, 3-4 x-10 x^{2},-2-5 x-6 x^{2}\right\\}\) (e) \(\left\\{1+2 x-x^{2}, 4-2 x+x^{2},-1+18 x-9 x^{2}\right\\}\)
Let $\mathrm{V}=\left\\{\left(a_{1}, a_{2}\right): a_{1}, a_{2} \in F\right\\}\(, where \)F\( is a field. Define addition of elements of \)\mathrm{V}$ coordinatewise, and for \(c \in F\) and $\left(a_{1}, a_{2}\right) \in \mathrm{V}$, define $$ c\left(a_{1}, a_{2}\right)=\left(a_{1}, 0\right) . $$ Is \(\mathrm{V}\) a vector space over \(F\) with these operations? Justify your answer.
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