Fall 2001  Math 2090  Final Exam Answers

1. [35 points] Consider the system of linear equations A x⃗ = b⃗:

\begin{aligned}x_{1} & + & 2x_{2} & + & x_{3} & = & 2 \\ 2x_{1} & + & 4x_{2} & + & 3x_{3} & = & 3 \\ x_{1} & + & 3x_{2} & + & x_{3} & = & 5 \end{aligned} where A = [ \begin{matrix}1 & 2 & 1 \\ 2 & 4 & 3 \\ 1 & 3 & 1 \end{matrix} ] and b⃗ = [ \begin{matrix}2 \\ 3 \\ 5 \end{matrix} ].

(a) Solve this (consistent) non-homogeneous system of linear equations.

(b) Is the matrix A nonsingular? Explain.

(c) Do the columns of the matrix A form a basis for ℝ^{3}? Explain.

(a) A^{♯} = [ \begin{matrix}1 & 2 & 1 & 2 \\ 2 & 4 & 3 & 3 \\ 1 & 3 & 1 & 5 \end{matrix} ] \begin{matrix}⟶ \\ R2 − 2R1 \\ R3 − R1 \end{matrix} [ \begin{matrix}1 & 2 & 1 & 2 \\ 0 & 0 & 1 & −1 \\ 0 & 1 & 0 & 3 \end{matrix} ] \overrightarrow{R2 ↔ R3} [ \begin{matrix}1 & 2 & 1 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & −1 \end{matrix} ] rank A = 3

The solution of A x⃗ = b⃗ is x⃗ = (x_{1}, x_{2}, x_{3}) = (−3, 3, −1)

(b) Since rank A = 3, the matrix A is nonsingular.

(c) Since rank A = 3, the system A x⃗ = 0 has only the trivial solution, and the system A x⃗ = b⃗ has a (unique) solution for any b⃗. Thus the columns of A are linearly independent, and span ℝ^{3}. So they do form a basis for ℝ^{3}.

2. [25 points] Consider the mapping T : ℝ^{3} → ℝ defined by T (x_{1}, x_{2}, x_{3}) = x_{1} − x_{3}.

(a) Verify that the mapping T is a linear transformation.

(b) Find a basis for Ker(T), the kernel of T, and find the dimension of Ker(T).

(a) T (u⃗ + v⃗) = T (u_{1} + v_{1}, u_{2} + v_{2}, u_{3} + v_{3}) = u_{1} + v_{1} − (u_{3} + v_{3}) = u_{1} − u_{3} + v_{1} − v_{3}

= T (u_{1}, u_{2}, u_{3}) + T (v_{1}, v_{2}, v_{3}) = T (u⃗) + T (v⃗)

T (k u⃗) = T (ku_{1}, ku_{2}, ku_{3}) = ku_{1} − ku_{3} = k(u_{1} − u_{3}) = k T (u_{1}, u_{2}, u_{3}) = k T (u⃗).

(b) Ker(T) = {x⃗ in ℝ^{3} | T (x⃗) = 0} = {x⃗ in ℝ^{3} | x_{1} − x_{3} = 0} = span{(1, 0, 1), (0, 1, 0)} basis: {(1, 0, 1), (0, 1, 0)} dim Ker(T) = 2

3. [40 points] Find the (general) solution of each of the following differential equations. Your answers should be real-valued.

(a) \frac{dy}{dx} + \frac{2}{x} y = \frac{6}{x}  (b) D(D + 1)^{2}(D^{2} + 9)y = 0  (c) y′′ − 2y′ − 3y = 6e^{2x}

(a) y = 3 + Cx^{−2}

(b) y = c_{1} + c_{2}e^{−x} + c_{3}xe^{−x} + c_{4} cos(3x) + c_{5} sin(3x)

(c) y = c_{1}e^{3x} + c_{2}e^{−x} − 2e^{2x}

Fall 2001  Math 2090  Final Exam Answers

4. [20 points] A table of Laplace transforms was included.

(a) Find the Laplace transform of the function f(t) = e^{2t} cos(3t) − 3e^{−2t}.

(b) Find the inverse Laplace transform of the function F (s) = \frac{6}{s^{3}} + \frac{s + 6}{s^{2} + 9}.

(a) L[e^{2t} cos(3t) − 3e^{−2t}] = \frac{s − 2}{(s − 2)^{2} + 9} − \frac{3}{s + 2} (b) L^{−1} [\frac{6}{s^{3}} + \frac{s + 6}{s^{2} + 9}] = 3t^{2} + cos(3t) + 2 sin(3t)

5. [20 points] Solve the initial value problem y′′ − 2y′ − 3y = 0, y(0) = 0, y′(0) = 8 two ways:

(a) using the Laplace transform  (b) without using the Laplace transform  y = 2e^{3t} − 2e^{−t}

6. [30 points] Consider the matrix A = [ \begin{matrix}2 & 0 & −1 \\ 1 & 2 & −2 \\ 1 & 0 & 0 \end{matrix} ].

(a) Verify that the characteristic polynomial of the matrix A is p(λ) = (1 − λ)^{2}(2 − λ).

(b) Find all eigenvalues and eigenvectors of A, and determine if A is defective or diagonalizable.

(a) p(λ) = det(A − λI) = | \begin{matrix}2 − λ & 0 & −1 \\ 1 & 2 − λ & −2 \\ 1 & 0 & 0 − λ \end{matrix} | = (2 − λ) | \begin{matrix}2 − λ & −1 \\ 1 & −λ \end{matrix} | = (2 − λ)(λ^{2} − 2λ + 1)

= (2 − λ)(1 − λ)^{2} (b) eigenvalues/eigenvectors: λ_{1} = 2, v⃗_{1} = (0, 1, 0), λ_{2} = λ_{3} = 1, v⃗_{2} = (1, 1, 1)

The matrix A is defective. λ = 1 has algebraic multiplicity two and geometric multiplicity one.

7. [30 points] The eigenvalues of a certain diagonalizable 2 × 2 matrix A are λ_{1} = 1 and λ_{2} = −1. Eigenvectors corresponding to λ_{1} and λ_{2} are v⃗_{1} = (2, 1) and v⃗_{2} = (1, 1).

(a) Find a fundamental set of solutions for the system of differential equations x⃗′ = A x⃗. Verify that the solutions you find form a fundamental set.

(b) Find a diagonal matrix D and a nonsingular matrix S so that S^{−1}AS = D.

(c) Find the matrix A.

(a) \{x⃗_{1}(t) = e^{t} [ \begin{matrix}2 \\ 1 \end{matrix}] , x⃗_{2}(t) = e^{−t} [ \begin{matrix}1 \\ 1 \end{matrix}]\} is a fundamental set of solutions W [x⃗_{1}, x⃗_{2}] = |\begin{matrix}2e^{t} & e^{−t} \\ e^{t} & e^{−t} \end{matrix}| = 1 ≠ 0

(b) D = [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] S = [ \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ] (c) A = SDS^{−1} = [ \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] [\begin{matrix}1 & −1 \\ −1 & 2 \end{matrix} ] = [ \begin{matrix}3 & −4 \\ 2 & −3 \end{matrix} ]

Extra Credit. For the matrix A from problem 7, compute A^{2001}.

A^{2001} = SD^{2001}S^{−1} = SDS^{−1} = A