MATH 54 1/31/2013 QUIZ SOLUTIONS
You have 15 minutes to complete the following quiz. Notice that there are two problems, one on
each side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more
sophisticated. It really isn
MATH 54 4/4 QUIZ
There is only one problem on this quiz.
1. Find all dierential functions y satisfying the following conditions:
y 4y + 4y = 0
y (0) = 1
y (1) = 0
Solution: The general solution to the given homogeneous equation is
ygen = Ae2x + Bxe2x .
Pl
MATH 54 4/11 QUIZ
1. Find a general solution to the following dierential equation:
y y = e2t + tet
1
2
MATH 54 4/11 QUIZ
2. Suppose that yp and yq are two solutions to the dierential equation
y + 2y + y = tan3 (x).
Suppose further that
yp (0) = yp (0) = 1
MATH 54 3/7 QUIZ
You have 20 minutes to complete this quiz. No notes or books are allowed.
1.
Let P2 be (as usual) the vector space of polynomials of degree 2. Let D : P2 P2 be the linear
transformation sending p(t) to p (t).
(a) Find the matrix of D rela
MATH 54 3/14 QUIZ
1. Which of the following are orthogonal matrices? Circle all that apply.
0 1/2
a.
0 1/ 2
b.
0 1
1 0
c.
1 2
2 1
d.
1 0
0 1
Solution: Only b and d are orthogonal matrices. A matrix has to have orthonormal columns to be
an orthogonal matri
MATH 54 MIDTERM 1 REVIEW
1. Make sure you review the denitions of the following terms.
(1) augmented matrix vs. coecient matrix
(2) echelon form and reduced echelon form
(3) pivots, pivot rows, pivot columns
(4) parametric form of solution set to Ax = b
(
MATH 54 2/7/2013 QUIZ
You have 15 minutes to complete the following quiz. Notice that there are problems on each
side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more
sophisticated. It really isnt necessary, though.
MATH 54 2/14/2013 DEFINITIONS QUIZ
1. Below is shown a matrix A, together with the reduced
[A|I].
1 4 2 1 0 0
1
[A|I] = 1 3 3 0 1 0 0
3 6 12 0 0 1
0
echelon form of the augmented matrix
0 6 3 4 0
1 1 1 1 0
0 0 3 6 1
Based on this calculation, answer the
MATH 54 QUIZ 2/28
1. Let P2 be the set of polynomials of degree 2. Find the change-of-basis matrix P from the
CB
basis B = cfw_1, t, t2 to the basis C = cfw_1 2t + t2 , 3 5t + 4t2 , 2t + 3t2 .
Solution: The matrix P has as columns the vectors [bi ]C . To
MATH 54 FINAL EXAM PRACTICE QUESTIONS
1. Let M be an m n matrix. In terms of the pivots of M , how do we tell if the matrix equation
M x = b always has at least one solution? At most one solution?
2. T/F: In some cases, it is possible for 4 vectors to spa