Math 524 Exam 1 Solutions
1. State the eight axioms of a vector space.
Vector addition is an abelian group. That is, addition is commutative, associative,
with an identity and inverses. More precisely:
x + y = y + x, x + (y + z) = (x + y) + z for all vect
Math 524 Exam 7 Solutions
1. Given any 2 2 matrix A, we consider the usual three systems, as below. If possible,
produce ve such matrices A, subject to the restrictions given. (I) x(n) = Ax(n 1),
(II) dx/dt = Ax, (III) d2 x/dt2 = Ax.
(a) (I) and (II) stab
Math 524 Exam 11 Solutions
0<x<1
Set s(x) = 1 otherwise . These problems all concern propagation of a square wave, with v = 1/3,
0
initial position f (x, 0) = f0 (x) = s(x), and initial velocity f (x, 0) = g0 (x) = 0. For each of the
t
following problems,
Math 524 Exam 10 Solutions
1. Find all 2 2 complex matrices that are simultaneously diagonal, Hermitian, and
unitary.
Let A = ( a 0 ), a diagonal matrix with complex entries. Its eigenvalues
0 b
are precisely a, b. Because A is Hermitian, they must be rea
Math 524 Exam 8 Solutions
All the problems concern the vector space R2 [t] and the bilinear real symmetric form f |g =
1
0
f (t)g(t)dt.
1. Under the standard basis E = cfw_1, t, t2 , nd the metric GE .
GE is the matrix satisfying f |g = [f ]T GE [g]E . Se
Math 524 Exam 6 Solutions
1/3 1/6
1/3 5/6
The rst three problems all concern A =
=
1/2 0
0 2/3
1 1
1 2
2 1
1 1
1. Solve the discrete-time system given by x(n) = Ax(n 1), with initial condition x(0) = ( 0 ).
1
A basis of eigenvectors is B = cfw_b1 , b2 , f
Math 524 Exam 5 Solutions
1. Suppose that A, B are square, diagonalizable matrices satisfying AB = BA + I. Without
using Thm. 4.10, prove that they are not simultaneously diagonalizable. (Note: Thm 4.10
says that A, B commute if and only if they are simul
Math 524 Exam 4 Solutions
For each of the following vector spaces V and linear operators L:
1. Find all eigenvalues.
2. Find a basis for each eigenspace.
3. Determine all algebraic and geometric multiplicities.
4. Is the operator diagonalizable?
A. V = R3
Math 524 Exam 4 Makeup Solutions
For each of the following vector spaces V and linear operators L:
1. Find all eigenvalues.
2. Find a basis for each eigenspace.
3. Determine all algebraic and geometric multiplicities.
4. Is the operator diagonalizable?
A.
Math 524 Exam 2 Solutions
All problems are for the vector space R2 [t], real polynomials of degree at most 2. We dene
V = cfw_p(t) : p(1) = 0, a subspace of R2 [t].
1. Let A = cfw_a1 , a2 for a1 = t 1, a2 = t2 1. Let B = cfw_b1 , b2 for b1 = t2 + t 2, b
Math 524 Exam 3 Solutions
Problems 1-4 are for the vector space R2 [t], real polynomials of degree at most 2. We dene
L : R2 [t] R2 [t] via L(f ) = (t 1) df .
dt
1. Directly calculate [L]E , for the basis E = cfw_1, t, t2 .
We again apply L to each basis