Chapter 6

a. Let \(p(x)\) and \(q(x)\) lie in \(\mathbf{P}_{1}\) and suppose that \(p(1) \neq 0, q(2) \neq 0,\) and \(p(2)=0=q(1) .\) Show that \(\\{p(x), q(x)\\}\) is a basis of \(\mathbf{P}_{1}\). [Hint: If \(r p(x)+s q(x)=0,\) evaluate at \(x=1, x=2 .]\) b. Let \(B=\left\\{p_{0}(x), p_{1}(x), \ldots, p_{n}(x)\right\\}\) be a set of polynomials in \(\mathbf{P}_{n}\). Assume that there exist numbers \(a_{0}, a_{1}, \ldots, a_{n}\) such that \(p_{i}\left(a_{i}\right) \neq 0\) for each \(i\) but \(p_{i}\left(a_{j}\right)=0\) if \(i\) is different from \(j .\) Show that \(B\) is a basis of \(\mathbf{P}_{n}\).

\(\mathbf{} \quad\) Let \(\\{\mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{z}\\}\) be independent. Which of the following are dependent? a. \(\\{\mathbf{u}-\mathbf{v}, \mathbf{v}-\mathbf{w}, \mathbf{w}-\mathbf{u}\\}\) b. \(\\{\mathbf{u}+\mathbf{v}, \mathbf{v}+\mathbf{w}, \mathbf{w}+\mathbf{u}\\}\) c. \(\\{\mathbf{u}-\mathbf{v}, \mathbf{v}-\mathbf{w}, \mathbf{w}-\mathbf{z}, \mathbf{z}-\mathbf{u}\\}\) d. \(\\{\mathbf{u}+\mathbf{v}, \mathbf{v}+\mathbf{w}, \mathbf{w}+\mathbf{z}, \mathbf{z}+\mathbf{u}\\}\)

Let \(V\) be the set of all infinite sequences \(\left(a_{0}, a_{1}, a_{2}, \ldots\right)\) of real numbers. Define addition and scalar multiplication by $$\left(a_{0}, a_{1}, \ldots\right)+\left(b_{0}, b_{1}, \ldots\right)=\left(a_{0}+b_{0}, a_{1}+b_{1}, \ldots\right)$$ and $$r\left(a_{0}, a_{1}, \ldots\right)=\left(r a_{0}, r a_{1}, \ldots\right)$$ a. Show that \(V\) is a vector space. b. Show that \(V\) is not finite dimensional. c. [For those with some calculus.] Show that the set of convergent sequences (that is, \(\lim _{n \rightarrow \infty} a_{n}\) exists) is a subspace, also of infinite dimension.

Let \(A\) be an \(n \times n\) matrix of rank \(r .\) If \(U=\left\\{X\right.\) in \(\left.\mathbf{M}_{n n} \mid A X=0\right\\},\) show that \(\operatorname{dim} U=n(n-r)\).

Let \(U\) and \(W\) be subspaces of \(V\) with bases \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\\}\) and \(\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}\right\\}\) respectively. If \(U\) and \(W\) have only the zero vector in common, show that \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{w}_{1}, \mathbf{w}_{2}\right\\}\) is independent.

Let \(A_{1}, A_{2}, \ldots, A_{m}\) denote \(n \times n\) matrices. If \(\mathbf{0} \neq \mathbf{y} \in \mathbb{R}^{n}\) and \(A_{1} \mathbf{y}=A_{2} \mathbf{y}=\cdots=A_{m} \mathbf{y}=\mathbf{0},\) show that \(\left\\{A_{1}, A_{2}, \ldots, A_{m}\right\\}\) cannot \(\operatorname{span} \mathbf{M}_{n n}\).

Let \(U\) and \(W\) be subspaces of \(V\). a. Show that \(U+W\) is a subspace of \(V\) containing both \(U\) and \(W\). b. Show that \(\operatorname{span}\\{\mathbf{u}, \mathbf{w}\\}=\mathbb{R} \mathbf{u}+\mathbb{R} \mathbf{w}\) for any vectors \(\mathbf{u}\) and \(\mathbf{w}\). c. Show that $$\begin{array}{l}\operatorname{span}\left\\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{m}, \mathbf{w}_{1}, \ldots,\mathbf{w}_{n}\right\\} \\ =\operatorname{span}\left\\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{m}\right\\}+\operatorname{span}\left\\{\mathbf{w}_{1}, \ldots, \mathbf{w}_{n}\right\\} \end{array}$$ for any vectors \(\mathbf{u}_{i}\) in \(U\) and \(\mathbf{w}_{j}\) in \(W\).

If \(U\) and \(W\) are subspaces of a vector space \(V\), let \(U \cup W=\\{\mathbf{v} \mid \mathbf{v}\) is in \(U\) or \(\mathbf{v}\) is in \(W\\}\). Show that \(U \cup W\) is a subspace if and only if \(U \subseteq W\) or \(W \subset U\).

If \(z\) is a complex number, show that \(\left\\{z, z^{2}\right\\}\) is independent if and only if \(z\) is not real.

If \(A\) and \(B\) are \(m \times n\) matrices, show that \(\operatorname{rank}(A+B) \leq \operatorname{rank} A+\operatorname{rank} B\).