MATH 54

Midterm 2 | Spring '16 | Yuan

1. (5 points) Find the rank of each of the following matrices. Explain your results briefly.
(a) The $4 \times 4$ matrix $\left( \begin{array}{c c c c} 1 & 0 & 0 & 4 \\ 0 & 2 & 0 & 5 \\ 0 & 0 & 3 & 6 \\ 0 & 0 & 1 & 2 \end{array} \right)$

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(b) The $5 \times 3$ matrix $\left( \begin{array}{c c c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \\ 13 & 14 & 15 \end{array} \right)$

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(c) For $a < b < c$, the $3 \times 3$ matrix $\left( \begin{array}{c c c} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{array} \right)$

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2. (5 points) Let $V$ be the vector space of all polynomials $f(t)$ of degree less than $5$ and satisfying $f(-t) = f(t)$. Let $T : V \to V$ be the linear transformation defined by $T(f(t)) = f(t) - f(0)$.
(Note: Below, we have assumed that $V$ is a vector space and that $T$ is a linear transformation. You are not required to prove these two statements, and you can use them freely.)
(a) Find the dimension of $V$.

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(b) Describe the kernel and the range of $T$.

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3. (5 points) Find all values of $a$ such that the matrix $A = \left( \begin{array}{c c c} 1 & 0 & 3 \\ 0 & 2 & 0 \\ 0 & 0 & a \end{array} \right)$ is diagonalizable.

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4. (5 points) Let $W$ be the subspace of $\mathbb{R}^4$ given by $x_1 + x_2 + x_3 + x_4 = 0$. Compute the orthogonal projection $\text{proj}_W (\vec{v})$ of the vector $\vec{v} = \left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right)$ to the space $W$.

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5. (5 points) Let $A$ be a $2 \times 2$ matrix which has two distinct real eigenvalues. Assume that $A^3 + A^2 - A - I_2 = 0_2$. Find all possible forms of the matrix $A^2$. (Note: $I_2$ denotes the $2 \times 2$ identity matrix, and $0_2$ denotes the $2 \times 2$ matrix whose entries are all $0$.)

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6. (5 points) Let $A$ and $B$ be $n \times n$ matrices, and assume that $B$ is invertible. Is it always true that $\text{rank}(AB) = \text{rank}(A)$? (Note: If your answer is yes, give a reason. If your answer is no, give a counter-example.)

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