Chapter 6

Which of the following are subspaces of \(\mathbf{M}_{22}\) ? Support your answer. a. \(U=\left\\{\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \mid a, b,\right.\) and \(c\) in \(\left.\mathbb{R}\right\\}\) b. \(U=\left\\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \mid a+b=c+d ; a, b, c, d\right.\) in \(\left.\mathbb{R}\right\\}\) c. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A=A^{T}\right\\}\) d. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A B=0\right\\}, B\) a fixed \(2 \times 2\) matrix e. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A^{2}=A\right\\}\) f. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A\right.\) is not invertible \(\\}\) g. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, B A C=C A B\right\\}, B\) and \(C\) fixed \(2 \times 2\) matrices

Are the following sets vector spaces with the indicated operations? If not, why not? a. The set \(V\) of nonnegative real numbers; ordinary addition and scalar multiplication. b. The set \(V\) of all polynomials of degree \(\geq 3\), together with 0 ; operations of \(\mathbf{P}\). c. The set of all polynomials of degree \(\leq 3\); operations of \(\mathbf{P}\). d. The set \(\left\\{1, x, x^{2}, \ldots\right\\} ;\) operations of \(\mathbf{P}\). e. The set \(V\) of all \(2 \times 2\) matrices of the form \(\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] ;\) operations of \(\mathbf{M}_{22}\) f. The set \(V\) of \(2 \times 2\) matrices with equal column sums; operations of \(\mathbf{M}_{22}\). g. The set \(V\) of \(2 \times 2\) matrices with zero determinant; usual matrix operations. h. The set \(V\) of real numbers; usual operations. i. The set \(V\) of complex numbers; usual addition and multiplication by a real number. j. The set \(V\) of all ordered pairs \((x, y)\) with the addition of \(\mathbb{R}^{2},\) but using scalar multiplication \(a(x, y)=(a x,-a y)\) \(\mathrm{k}\). The set \(V\) of all ordered pairs \((x, y)\) with the addition of \(\mathbb{R}^{2}\), but using scalar multiplication \(a(x, y)=(x, y)\) for all \(a\) in \(\mathbb{R}\) 1\. The set \(V\) of all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) with pointwise addition, but scalar multiplication defined by \((a f)(x)=f(a x)\) \(\mathrm{m}\). The set \(V\) of all \(2 \times 2\) matrices whose entries sum to \(0 ;\) operations of \(\mathbf{M}_{22}\). n. The set \(V\) of all \(2 \times 2\) matrices with the addition of \(\mathbf{M}_{22}\) but scalar multiplication \(*\) defined by \(a * X=a X^{T}\).

Prove Taylor's theorem for polynomials.

Which of the following are subspaces of \(\mathbf{F}[0,1] ?\) Support your answer. a. \(U=\\{f \mid f(0)=0\\}\) b. \(U=\\{f \mid f(0)=1\\}\) c. \(U=\\{f \mid f(0)=f(1)\\}\) d. \(U=\\{f \mid f(x) \geq 0\) for all \(x\) in [0,1]\(\\}\) e. \(U=\\{f \mid f(x)=f(y)\) for all \(x\) and \(y\) in [0,1]\(\\}\) f. \(U=\\{f \mid f(x+y)=f(x)+f(y)\) for all \(x\) and \(y\) in [0,1]\(\\}\) g. \(U=\left\\{f \mid f\right.\) is integrable and \(\left.\int_{0}^{1} f(x) d x=0\right\\}\)

In each case, find a basis of \(V\) containing \(\mathbf{v}\) and \(\mathbf{w}\) a. \(V=\mathbb{R}^{4}, \mathbf{v}=(1,-1,1,-1), \mathbf{w}=(0,1,0,1)\) b. \(V=\mathbb{R}^{4}, \mathbf{v}=(0,0,1,1), \mathbf{w}=(1,1,1,1)\) c. \(V=\mathbf{M}_{22}, \mathbf{v}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], \mathbf{w}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) d. \(V=\mathbf{P}_{3}, \mathbf{v}=x^{2}+1, \mathbf{w}=x^{2}+x\)

Use Taylor's theorem to derive the binomial theorem: $$(1+x)^{n}=\left(\begin{array}{l}n \\\0\end{array}\right)+\left(\begin{array}{l}n \\\1\end{array}\right) x+\left(\begin{array}{l}n \\\2\end{array}\right) x^{2}+\cdots+\left(\begin{array}{l}n \\\n\end{array}\right) x^{n}$$\ Here the binomial coefficients \(\left(\begin{array}{c}n \\\ r\end{array}\right)\) are defined by $$\left(\begin{array}{l}n \\\r\end{array}\right)=\frac{n !}{r !(n-r) !}$$ where \(n !=n(n-1) \cdots 2 \cdot 1\) if \(n \geq 1\) and \(0 !=1\).

a. Given the equation \(f^{\prime}+a f=b,(a \neq 0),\) make the substitution \(f(x)=g(x)+b / a\) and obtain a differential equation for \(g\). Then derive the general solution for \(f^{\prime}+a f=b\). b. Find the general solution to \(f^{\prime}+f=2\).

Let \(A\) be an \(m \times n\) matrix. For which columns b in \(\mathbb{R}^{m}\) is \(U=\left\\{\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^{n}, A \mathbf{x}=\mathbf{b}\right\\}\) a subspace of \(\mathbb{R}^{n} ?\) Support your answer.

a. If \(z\) is not a real number, show that \(\left\\{z, z^{2}\right\\}\) is a basis of the real vector space \(\mathbb{C}\) of all complex numbers. b. If \(z\) is neither real nor pure imaginary, show that \(\\{z, \bar{z}\\}\) is a basis of \(\mathbb{C}\)

Find all values of \(a\) such that the following are independent in \(\mathbb{R}^{3}\). $$\text { a. }\\{(1,-1,0),(a, 1,0),(0,2,3)\\}$$ b. \(\\{(2, a, 1),(1,0,1),(0,1,3)\\}\)