Middle East Technical University Basic Linear Algebra Math 260 Spring 2005

Make-up Exam

1. (3+3+3 pts) Let P_{2} = {a_{2}x^{2} + a_{1}x + a_{0}} be the space of polynomials of degree ⩽ 2, and L: P_{2} → P_{2} is the linear map, p(x) ↦ p′(x) − xp′′(x).

a) Determine the matrix of L with respect to the basis {1, x, x^{2}}.

b) Determine which polynomials p(x) belong to the kernel (null-space) of L.

c) Determine all those p ∈ P_{2} such that L(p) = x + 1.

2. (3+3+3 pts) Let (f, g) = 3f(0)g(0) + f′(0)g′(0) + f′′(0)g′′(0).

a) Prove that this is an inner product in the space P_{2} from Problem 1.

b) Explain why this is not an inner product in the space P_{3} = {a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}} of polynomials of degree ⩽ 3.

c) Find an orthogonal basis of P_{2} with respect to this inner product

3. (4+3+4 pts) Consider a linear map L: ℝ^{2×2} → ℝ^{3} from the space of 2 × 2

matrices ℝ^{2×2} to ℝ^{3}, defined by the matrix ( \begin{matrix}1 & 2 & 3 & 4 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} ) with respect to the

(ordered) bases A_{1} = ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ), A_{2} = ( \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ), A_{3} = ( \begin{matrix}1 & 2 \\ 0 & 0 \end{matrix} ), A_{4} = ( \begin{matrix}0 & 1 \\ 0 & 1 \end{matrix} ) in

ℝ^{2×2} and e_{1} = (1, 0, 1), e_{2} = (0, 1, 1), and e_{3} = (1, 1, 1) in ℝ^{3}.

a) Determine L ( \begin{matrix}a & b \\ c & d \end{matrix} )

b) Determine the 2 × 2 matrices which belong to the kernel of L.

c) Find the matrix of L with respect to the bases A_{2}, A_{4}, A_{1}, A_{3} and e_{3}, e_{2}, e_{1}.

4. (3+4+4 pts) Let A be 3 × 3 matrix such that Av_{1} = −v_{1} Av_{2} = 2v_{1}

Av_{3} = 2v_{3} for v_{1} = ( \begin{matrix}1 \\ 0 \\ 2 \end{matrix} ), v_{2} = ( \begin{matrix}1 \\ 1 \\ 0 \end{matrix} ), v_{3} = ( \begin{matrix}1 \\ 2 \\ 1 \end{matrix} ).

a) Determine the trace and the determinant of A.

b) Determine the characteristic polynomial of matrix B = A^{3}.

c) Determine matrix C = A^{2}.