91Ó°ÊÓ

Show that \(P_{n}(F)\) is generated by \(\left\\{1, x, \ldots, x^{n}\right\\}\).

Let \(W\) denote the subspace of \(R^{5}\) consisting of all the vectors having coordinates that sum to zero. The vectors $$ \begin{array}{ll} u_{1}=(2,-3,4,-5,2), & u_{2}=(-6,9,-12,15,-6), \\ u_{3}=(3,-2,7,-9,1), & u_{4}=(2,-8,2,-2,6), \\ u_{5}=(-1,1,2,1,-3), & u_{6}=(0,-3,-18,9,12), \\ u_{7}=(1,0,-2,3,-2), & u_{8}=(2,-1,1,-9,7) \end{array} $$ generate W. Find a subset of the set \(\left\\{u_{1}, u_{2}, \ldots, u_{8}\right\\}\) that is a basis for W.

Let \(S=\\{(1,1,0),(1,0,1),(0,1,1)\\}\) be a subset of the vector space \(\mathrm{F}^{3}\). (a) Prove that if \(F=R\), then \(S\) is linearly independent. (b) Prove that if \(F\) has characteristic two, then \(S\) is linearly dependent.

Determine whether the following sets are subspaces of \(\mathrm{R}^{3}\) under the operations of addition and scalar multiplication defined on \(R^{3}\). Justify your answers. (a) \(\mathrm{W}_{1}=\left\\{\left(a_{1}, a_{2}, a_{3}\right) \in \mathrm{R}^{3}: a_{1}=3 a_{2}\right.\) and \(\left.a_{3}=-a_{2}\right\\}\) (b) \(\mathrm{W}_{2}=\left\\{\left(a_{1}, a_{2}, a_{3}\right) \in \mathrm{R}^{3}: a_{1}=a_{3}+2\right\\}\) (c) \(\mathbf{W}_{3}=\left\\{\left(a_{1}, a_{2}, a_{3}\right) \in \mathbf{R}^{3}: 2 a_{1}-7 a_{2}+a_{3}=0\right\\}\) (d) \(\mathbf{W}_{4}=\left\\{\left(a_{1}, a_{2}, a_{3}\right) \in \mathbf{R}^{3}: a_{1}-4 a_{2}-a_{3}=0\right\\}\) (e) \(\mathrm{W}_{5}=\left\\{\left(a_{1}, a_{2}, a_{3}\right) \in \mathrm{R}^{3}: a_{1}+2 a_{2}-3 a_{3}=1\right\\}\) (f) \(\mathrm{W}_{6}=\left\\{\left(a_{1}, a_{2}, a_{3}\right) \in \mathrm{R}^{3}: 5 a_{1}^{2}-3 a_{2}^{2}+6 a_{3}^{2}=0\right\\}\)

The vectors \(u_{1}=(1,1,1,1), u_{2}=(0,1,1,1), u_{3}=(0,0,1,1)\), and \(u_{4}=(0,0,0,1)\) form a basis for \(F^{4}\). Find the unique representation of an arbitrary vector \(\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\) in \(\mathrm{F}^{4}\) as a linear combination of \(u_{1}, u_{2}, u_{3}\), and \(u_{4}\).

Show that the matrices $$ \left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right) $$ generate \(\mathrm{M}_{2 \times 2}(F)\).

Let \(V\) denote the set of all differentiable real-valued functions defined on the real line. Prove that \(V\) is a vector space with the operations of addition and scalar multiplication defined in Example \(3 .\)

Let \(V=\\{0\\}\) consist of a single vector 0 and define \(0+0=0\) and \(c 0=0\) for each scalar \(c\) in \(F\). Prove that \(\mathrm{V}\) is a vector space over \(F\). ( \(V\) is called the zero vector space.)

Prove that \(\operatorname{span}(\\{x\\})=\\{a x: a \in F\\}\) for any vector \(x\) in a vector space, Interpret this result geometrically in \(\mathrm{R}^{3}\).

A real-valued function \(f\) defined on the real line is called an even function if \(f(-t)=f(t)\) for each real number \(t\). Prove that the set of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space.