Queens University Math 110 Exam #2 April 2022 PDF

# QUEEN'S UNIVERSITY ## FACULTY OF ARTS AND SCIENCE ## DEPARTMENT OF MATHEMATICS & STATISTICS ### MATH 110 ### EXAM #2 ### APRIL 2022 - This examination is three hours in length. - Calculators, data sheets, or other aids are not permitted. - Each question is worth 10 points. - To receive full cred...

# QUEEN'S UNIVERSITY ## FACULTY OF ARTS AND SCIENCE ## DEPARTMENT OF MATHEMATICS & STATISTICS ### MATH 110 ### EXAM #2 ### APRIL 2022 - This examination is three hours in length. - Calculators, data sheets, or other aids are not permitted. - Each question is worth 10 points. - To receive full credit, you must explain your answers. - Answers are to be recorded on the question paper. - Proctors are unable to respond to queries about the interpretations of exam questions. Do your best to answer exam questions as written. This material is copyrighted and is for the sole use of students registered in Math110 and writing this examination. This material shall not be distributed or disseminated. Failure to abide by these conditions is a breach of copyright and may also constitute a breach of academic integrity under the University Senate's Academic Integrity Policy Statement. # 1. Let $\mathbb{R}^{\mathbb{R}}$ be the real vector space of all real-valued functions on the real line. - Prove that the set $U$ of all function $f$ in $\mathbb{R}^{\mathbb{R}}$ that vanish at the integer multiples of $2\pi$ forms a linear subspace; $U := \{ f \in \mathbb{R}^{\mathbb{R}} \mid f(2k\pi) = 0$ for all $k\in \mathbb{Z} \}$. - Prove that the three functions $x^2-1$, $sin(x)$, and $exp(x^2)$ are linearly independent in $\mathbb{R}^{\mathbb{R}}$. # 2. Let $\mathbb{Q}^{2 \times 2}$ denote the rational vector space of all rational $(2 \times 2)$-matrices. Consider the linear operator $\Phi: \mathbb{Q}^{2 \times 2} \to \mathbb{Q}^{2 \times 2}$ defined, for all a, b, c, and d in $\mathbb{Q}$, by $\Phi \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d-a & 0 \\ b+ a-c & a-c \end{pmatrix}$. - Let $E_{j,k}$ be the $(2 \times 2)$-matrix having 1 in the $(j,k)$-entry and 0 elsewhere. Find the matrix of the linear operator $\Phi$ relative to the ordered basis $\epsilon = (E_{1,1}, E_{2,1}, E_{1,2}, E_{2,2})$. - Is the linear operator $\Phi$ invertible? Explain why or why not. # 3. Let $\mathbb{C}[t]_{<2}$ be the complex vector space of all polynomials in the variable $t$ having degree at most 2. Consider the the linear operator $\Psi: \mathbb{C}[t]_{<2} \to \mathbb{C}[t]_{<2}$ defined, for all polynomials $q$ in $\mathbb{C}[t]_{<2}$, by $\Psi[q] = \frac{q(t)-q(1)}{t-1} + q(0) + \frac{1}{2}(q(1) + 2q(0) - q(-1))t^2$. - Find the matrix of the linear operator $\Psi$ relative to the monomial basis $M = (1, t, t^2)$. - Is the linear operator $\Psi$ diagonalizable? Explain why or why not. # 4. The eigenvalues of the matrix $M = \begin{pmatrix} -3 & -1 & 1 \\ -2 & 5 & -4 \\ -3 & 6 & -5 \end{pmatrix}$ are -2 and 1. - Find a basis for each eigenspace of $M$. - Is the matrix $M$ diagonalizable? Explain why or why not. # 5. For any three vectors $u$, $v$, and $w$ in a complex inner product space, prove that $\frac{||w - u||^2 + ||w - v||^2}{2} - \frac{||u - v||^2}{4} = ||\frac{1}{2}(u + v) - w||^2$. # 6. Equip the coordinate space $\mathbb{R}^3$ with the weighted (or non-standard) inner product defined, for all vectors $v := \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$ and $w := \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}$ in $\mathbb{R}^3$, by $(v, w) = (v_1w_1+2v_2w_2+3v_3w_3)$. - Verify that the vectors $z_1 := \begin{pmatrix} 3 \\ 3 \\ -3 \end{pmatrix}$ and $z_2 := \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$ are orthogonal. - Calculate the norms of the vectors $z_1 := \begin{pmatrix} 3 \\ 3 \\ -3 \end{pmatrix}$ and $z_2 := \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$. - Compute the orthogonal projection of $\begin{pmatrix} -3 \\ 6 \\ -1 \end{pmatrix}$ onto the linear subspace $Z := Span(z_1, z_2)$. # 7. Find a vector $x$ in $\mathbb{R}^3$ such that $||Ax - b||$ is minimal where $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & 1 & 2 \end{pmatrix}$ and $b = \begin{pmatrix} 4 \\ 0 \\ 0 \end{pmatrix}$. # 8. Let $W$ be finite-dimensional complex inner product space and let $S: W \to W$ be an invertible linear operator. Show that $S^{-1}S^*$ is an isometry if and only if $S$ is normal. # 9. Let $V$ be a finite-dimensional complex inner product space and let $T: V \to V$ be a normal linear operator such that $T^6 = T^5$. Prove that $T$ is self-adjoint and $T^2 = T$. # 10. Find a singular-value decomposition of the matrix $B = \begin{pmatrix} 2 & 2 \\ -1 & 2 \end{pmatrix}$.