91Ó°ÊÓ

Let \(F_{1}\) and \(F_{2}\) be fields. A function \(g \in \mathcal{F}\left(F_{1}, F_{2}\right)\) is called an even function if \(g(-t)=g(t)\) for each \(t \in F_{1}\) and is called an odd function if \(g(-t)=-g(t)\) for each \(t \in F_{1}\). Prove that the set of all even functions in \(\mathcal{F}\left(F_{1}, F_{2}\right)\) and the set of all odd functions in \(\mathcal{F}\left(F_{1}, F_{2}\right)\) are subspaces of \(\mathcal{F}\left(F_{1}, F_{2}\right)\).

Let \(v_{1}, v_{2}, \ldots, v_{k}, v\) be vectors in a vector space \(\mathrm{V}\), and define \(\mathrm{W}_{1}=\) \(\operatorname{span}\left(\left\\{v_{1}, v_{2}, \ldots, v_{k}\right\\}\right)\), and \(\mathrm{W}_{2}=\operatorname{span}\left(\left\\{v_{1}, v_{2}, \ldots, v_{k}, v\right\\}\right)\). (a) Find necessary and sufficient conditions on \(v\) such that \(\operatorname{dim}\left(\mathrm{W}_{1}\right)=\) \(\operatorname{dim}\left(\mathrm{W}_{2}\right)\). (b) State and prove a relationship involving \(\operatorname{dim}\left(\mathrm{W}_{1}\right)\) and \(\operatorname{dim}\left(\mathrm{W}_{2}\right)\) in the case that \(\operatorname{dim}\left(\mathrm{W}_{1}\right) \neq \operatorname{dim}\left(\mathrm{W}_{2}\right)\).

Show that \(F^{n}\) is the direct sum of the subspaces $$ \mathrm{W}_{1}=\left\\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathrm{F}^{n}: a_{n}=0\right\\} $$ and $$ \mathrm{W}_{2}=\left\\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathrm{F}^{n}: a_{1}=a_{2}=\cdots=a_{n-1}=0\right\\} . $$

Let \(\mathrm{W}_{1}\) denote the set of all polynomials \(f(x)\) in \(\mathrm{P}(F)\) such that in the representation $$ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, $$ we have \(a_{i}=0\) whenever \(i\) is even. Likewise let \(\mathrm{W}_{2}\) denote the set of all polynomials \(g(x)\) in \(\mathrm{P}(F)\) such that in the representation $$ g(x)=b_{m} x^{m}+b_{m-1} x^{m-1}+\cdots+b_{1} x+b_{0}, $$ we have \(b_{i}=0\) whenever \(i\) is odd. Prove that \(\mathrm{P}(F)=\mathrm{W}_{1} \oplus \mathrm{W}_{2}\).

For a fixed \(a \in R\), determine the dimension of the subspace of \(\mathrm{P}_{n}(R)\) defined by \(\left\\{f \in \mathrm{P}_{n}(R): f(a)=0\right\\}\).

Let \(V\) denote the vector space of all upper triangular \(n \times n\) matrices (as defined on page 19), and let \(W_{1}\) denote the subspace of \(V\) consisting of all diagonal matrices. Define \(\mathrm{W}_{2}=\left\\{A \in \mathrm{V}: A_{i j}=0\right.\) whenever \(\left.i \geq j\right\\}\). Show that \(\mathrm{V}=\mathrm{W}_{1} \oplus \mathrm{W}_{2}\).

Let \(V\) be a finite-dimensional vector space over \(C\) with dimension \(n\). Prove that if \(\mathrm{V}\) is now regarded as a vector space over \(R\), then \(\operatorname{dim} \mathrm{V}=\) \(2 n\). (See Examples 11 and 12.)

A matrix \(M\) is called skew-symmetric if \(M^{t}=-M .\) Clearly, a skew- symmetric matrix is square. Let \(F\) be a field. Prove that the set W \(_{1}\) of all skew-symmetric \(n \times n\) matrices with entries from \(F\) is a sub- space of \(M_{n \times n}(F) .\) Now assume that \(F\) is not of characteristic two (see page 549\(),\) and let \(W_{2}\) be the subspace of \(M_{n \times n}(F)\) consisting of all symmetric \(n \times n\) matrices. Prove that \(M_{n \times n}(F)=W_{1} \oplus W_{2}\).

Let \(F\) be a field that is not of characteristic two. Define $$ \mathrm{W}_{1}=\left\\{A \in \mathrm{M}_{n \times n}(F): A_{i j}=0 \text { whenever } i \leq j\right\\} $$ and \(\mathrm{W}_{2}\) to be the set of all symmetric \(n \times n\) matrices with entries from \(F\). Both \(\mathrm{W}_{1}\) and \(\mathrm{W}_{2}\) are subspaces of \(\mathrm{M}_{n \times n}(F)\). Prove that \(\mathrm{M}_{n \times n}(F)=\mathrm{W}_{1} \oplus \mathrm{W}_{2}\). Compare this exercise with Exercise 28 .

Let \(W_{1}\) and \(W_{2}\) be subspaces of a vector space \(V\). Prove that \(V\) is the direct sum of \(W_{1}\) and \(W_{2}\) if and only if each vector in \(V\) can be uniquely written as \(x_{1}+x_{2}\), where \(x_{1} \in \mathrm{W}_{1}\) and \(x_{2} \in \mathrm{W}_{2}\).