Problem 30
Show that \(\left\\{(a, b),\left(a_{1}, b_{1}\right)\right\\}\) is a basis of \(\mathbb{R}^{2}\) if and only if \(\left\\{a+b x, a_{1}+b_{1} x\right\\}\) is a basis of \(\mathbf{P}_{1}\).
Problem 31
Find the dimension of the subspace \(\operatorname{span}\left\\{1, \sin ^{2} \theta, \cos 2 \theta\right\\}\) of \(\mathbf{F}[0,2 \pi]\).
Problem 33
If \(U\) and \(W\) are subspaces of \(V,\) define their intersection \(U \cap W\) as follows: \(U \cap W=\\{\mathbf{v} \mid \mathbf{v}\) is in both \(U\) and \(W\\}\) a. Show that \(U \cap W\) is a subspace contained in \(U\) and \(W\). b. Show that \(U \cap W=\\{\mathbf{0}\\}\) if and only if \(\\{\mathbf{u}, \mathbf{w}\\}\) is independent for any nonzero vectors \(\mathbf{u}\) in \(U\) and \(\mathbf{w}\) in \(W\). c. If \(B\) and \(D\) are bases of \(U\) and \(W,\) and if \(U \cap W=\) \(\\{\boldsymbol{0}\\},\) show that \(B \cup D=\\{\mathbf{v} \mid \mathbf{v}\) is in \(B\) or \(D\\}\) is independent.
Problem 35
Let \(D_{n}\) denote the set of all functions \(f\) from the set \(\\{1,2, \ldots, n\\}\) to \(\mathbb{R}\). a. Show that \(\mathbf{D}_{n}\) is a vector space with pointwise addition and scalar multiplication. b. Show that \(\left\\{S_{1}, S_{2}, \ldots, S_{n}\right\\}\) is a basis of \(\mathbf{D}_{n}\) where, for each \(k=1,2, \ldots, n,\) the function \(S_{k}\) is defined by \(S_{k}(k)=1,\) whereas \(S_{k}(j)=0\) if \(j \neq k\).
Problem 36
A polynomial \(p(x)\) is called even if \(p(-x)=p(x)\) and odd if \(p(-x)=-p(x)\). Let \(E_{n}\) and \(O_{n}\) denote the sets of even and odd polynomials in \(\mathbf{P}_{n}\). a. Show that \(E_{n}\) is a subspace of \(\mathbf{P}_{n}\) and find \(\operatorname{dim} E_{n}\). b. Show that \(O_{n}\) is a subspace of \(\mathbf{P}_{n}\) and find \(\operatorname{dim} O_{n}\).
