Problem 1
If polynomials \(f(x)\) and \(g(x)\) satisfy \(f(a)=g(a),\) show that \(f(x)-g(x)=(x-a) h(x)\) for some polynomial \(h(x)\).
Problem 1
In each case, find a basis for \(V\) that includes the vector \(\mathbf{v}\). a. \(V=\mathbb{R}^{3}, \mathbf{v}=(1,-1,1)\) b. \(V=\mathbb{R}^{3}, \mathbf{v}=(0,1,1)\) c. \(V=\mathbf{M}_{22}, \mathbf{v}=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) d. \(V=\mathbf{P}_{2}, \mathbf{v}=x^{2}-x+1\)
Problem 1
Show that each of the following sets of vectors is independent. a. \(\left\\{1+x, 1-x, x+x^{2}\right\\}\) in \(\mathbf{P}_{2}\) b. \(\left\\{x^{2}, x+1,1-x-x^{2}\right\\}\) in \(\mathbf{P}_{2}\) \(\left\\{\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{rr}0 & 0 \\ 1 & -1\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right]\right\\}\) \(\left\\{\left[\begin{array}{ll}1 & 1 \\ 1 & 0 \\ \text { in } & \mathbf{M}_{22}\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\right\\}\)
Problem 1
Let \(V\) denote the set of ordered triples \((x, y, z)\) and define addition in \(V\) as in \(\mathbb{R}^{3}\). For each of the following definitions of scalar multiplication, decide whether \(V\) is a vector space. a. \(a(x, y, z)=(a x, y, a z)\) b. \(a(x, y, z)=(a x, 0, a z)\) c. \(a(x, y, z)=(0,0,0)\) d. \(a(x, y, z)=(2 a x, 2 a y, 2 a z)\)
Problem 1
Which of the following are subspaces of \(\mathbf{P}_{3}\) ? Support your answer. a. \(U=\left\\{f(x) \mid f(x) \in \mathbf{P}_{3}, f(2)=1\right\\}\) b. \(U=\left\\{x g(x) \mid g(x) \in \mathbf{P}_{2}\right\\}\) c. \(U=\left\\{x g(x) \mid g(x) \in \mathbf{P}_{3}\right\\}\) d. \(U=\left\\{x g(x)+(1-x) h(x) \mid g(x)\right.\) and \(\left.h(x) \in \mathbf{P}_{2}\right\\}\) e. \(U=\) The set of all polynomials in \(\mathbf{P}_{3}\) with constant term 0 f. \(U=\left\\{f(x) \mid f(x) \in \mathbf{P}_{3},\right.\) deg \(\left.f(x)=3\right\\}\)
Problem 1
Find a solution \(f\) to each of the following differential equations satisfying the given boundary conditions. a. \(f^{\prime}-3 f=0 ; f(1)=2\) b. \(f^{\prime}+f=0 ; f(1)=1\) c. \(f^{\prime \prime}+2 f^{\prime}-15 f=0 ; f(1)=f(0)=0\) d. \(f^{\prime \prime}+f^{\prime}-6 f=0 ; f(0)=0, f(1)=1\) e. \(f^{\prime \prime}-2 f^{\prime}+f=0 ; f(1)=f(0)=1\) f. \(f^{\prime \prime}-4 f^{\prime}+4 f=0 ; f(0)=2, f(-1)=0\) g. \(f^{\prime \prime}-3 a f^{\prime}+2 a^{2} f=0 ; a \neq 0 ; f(0)=0,\) \(f(1)=1-e^{a}\) h. \(f^{\prime \prime}-a^{2} f=0, a \neq 0 ; f(0)=1, f(1)=0\) i. \(f^{\prime \prime}-2 f^{\prime}+5 f=0 ; f(0)=1, f\left(\frac{\pi}{4}\right)=0\) j. \(f^{\prime \prime}+4 f^{\prime}+5 f=0 ; f(0)=0, f\left(\frac{\pi}{2}\right)=1\)
Problem 2
If the characteristic polynomial of \(f^{\prime \prime}+a f^{\prime}+b f=0\) has real roots, show that \(f=0\) is the only solution satisfying \(f(0)=0=f(1)\).
Problem 2
In each case, find a basis for \(V\) among the given vectors. $$\begin{array}{l}\text { a. } V=\mathbb{R}^{3}, \\\\\quad\\{(1,1,-1),(2,0,1),(-1,1,-2),(1,2,1)\\}\end{array}$$ b. \(V=\mathbf{P}_{2},\left\\{x^{2}+3, x+2, x^{2}-2 x-1, x^{2}+x\right\\}\)
Problem 2
Which of the following subsets of \(V\) are independent? a. \(V=\mathbf{P}_{2} ;\left\\{x^{2}+1, x+1, x\right\\}\) b. \(V=\mathbf{P}_{2} ;\left\\{x^{2}-x+3,2 x^{2}+x+5, x^{2}+5 x+1\right\\}\) c. \(V=\mathbf{M}_{22} ;\left\\{\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\right\\}\) d. \(V=\mathbf{M}_{22}\) \(\left\\{\left[\begin{array}{rr}-1 & 0 \\ 0 & -1\end{array}\right],\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right],\left[\begin{array}{rr}0 & -1 \\ -1 & 0\end{array}\right]\right\\}\) e. \(V=\mathbf{F}[1,2] ;\left\\{\frac{1}{x}, \frac{1}{x^{2}}, \frac{1}{x^{3}}\right\\}\) f. \(V=\mathbf{F}[0,1] ;\left\\{\frac{1}{x^{2}+x-6}, \frac{1}{x^{2}-5 x+6}, \frac{1}{x^{2}-9}\right\\}\)
Problem 2
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
