Math 250B
Group Work #6
SOLUTIONS
Fall 2013
Problem 1. Suppose A is an n n matrix such that A8 = 0n . Show that the matrix
In + A2 is invertible and
(In + A2 )1 = In A2 + A4 A6 .
SOLUTION: To check if two things are inverses of each other, we multiply the
September 11, 2013
Quiz 1
Math 250B
Name:
Problem 1. (3 points each) Decide whether each dierential equation below is separable,
linear, Bernoulli, or homogeneous, and circle your answer:
(a)
y
SEPARABLE
y
= 2x2 ln x
x
LINEAR
BERNOULLI
HOMOGENEOUS
(b)
y
Math 250B
Midterm III Information
Fall 2013
SOLUTIONS TO PRACTICE PROBLEMS
Problem 1. Determine whether the following matrix is diagonalizable or
not. If it is, nd an invertible matrix S and a diagonal matrix D such that
S 1 AS = D.
2
0
A :=
0
0
0
3
0
September 11, 2013
Quiz 1
Math 250B
SOLUTIONS
Name:
Problem 1. (3 points each) Decide whether each dierential equation below is
separable, linear, Bernoulli, or homogeneous, and circle your answer:
(a)
y
y
= 2x2 ln x
x
2
SOLUTION: LINEAR
(b)
y [ln(y/x) +
Math 250B
Midterm III Information
Spring 2014
SOLUTIONS TO PRACTICE PROBLEMS
Problem 1. Determine whether the following matrix is diagonalizable or
not. If it is, nd an invertible matrix S and a diagonal matrix D such that
S 1 AS = D.
2
0
A :=
0
0
0
3
October 23, 2013
Quiz 2
Math 250B
Name:
Problem 1. (6 points) Does
S :=
1 1
0 1
,
1 3
1 0
,
1 0
1 2
form a spanning set for M2 (R) and is this set linearly independent? CIRCLE THE BEST
ANSWER and show work and explain why you circled the answer you did. N
October 23, 2013
Quiz 2
Math 250B
SOLUTIONS
Name:
Problem 1. (6 points) Does
1 1
0 1
S :=
,
1 3
1 0
,
1 0
1 2
form a spanning set for M2 (R) and is this set linearly independent? CIRCLE
THE BEST ANSWER and show work and explain why you circled the answer
December 2013
Midterm III Review Session
Math 250B
Name:
Problem 1. Suppose A is a square matrix such that
A3 2A2 + 4A + 5I = 0.
If is an eigenvalue of A, show that
3 22 + 4 + 5 = 0.
Problem 2. Suppose A is a square matrix with eigenvalues 1 = 2 . Show th
December 4, 2013
Quiz 3
Math 250B
Name:
Problem 1. (8 points) Decide whether the matrix A below is diagonalizable or not, circle
your answer, and show work to justify it. If the matrix is diagaonalizable, nd an invertible
matrix S and a diagonal matrix D
Math 250B
Midterm I Review Session
Spring 2014
Approximate Exam Breakdown:
Chapter 1: 40%
Chapter 2: 35%
Chapter 3: 25%
Ten Good Formulas to Know:
(1): Integrating factor: I(x) = e
p(x)dx
(2): u = y 1n converts Bernoulli equation to u + (1 n)p(x)u = (1 n)
December 4, 2013
Quiz 3
Math 250B
SOLUTIONS
Name:
Problem 1. (8 points) Decide whether the matrix A below is diagonalizable
or not, circle your answer, and show work to justify it. If the matrix is diagaonalizable, nd an invertible matrix S and a diagonal
Math 250B
Midterm I Practice Problems
SOLUTIONS
Spring 2014
Problem 1. Find the inverse of each matrix below, if it is invertible, and
express each invertible matrix as a product of elementary matrices:
3 4 1
3
(a): 1 0
2 5 4
SOLUTION: These are routine
Math 250B
Midterm III Information
Spring 2014
WHEN: Monday, May 5, in class (no notes, books, calculators I will supply a table
of annihilators and integrals.)
EXTRA OFFICE HOURS
Friday, May 2
10:3012:00 NOON and 4:00 6:00 PM and 9:00 11:00 PM p.m.
Saturd
Math 250B
Group Work #7
SOLUTIONS
Fall 2013
Problem 1. Let V = M2 (R) and let S be the subset of V consisting of all 2 2 matrices
whose four entries add up to zero.
(a): Show that S is a subspace of V .
a
b
. Since S will inherit
c (a + b + c)
most of the
Math 250B
Group Work #8
SOLUTIONS
Fall 2013
Problem 1. Let V = M23 (R) and let W be the subset of V consisting of all 2 3 matrices
whose rows and columns each sum to zero. Find a set of vectors that span W .
SOLUTION: If A W , then we can write
a
a
A=
a b
Math 250B
Group Work #9
SOLUTIONS
Fall 2013
Problem 1. Find a basis for the subspace S of P3 (R) consisting of all polynomials of the
form ax3 + (a + 3b + c)x2 + (b + c)x 3a. That is, nd a basis for
S = cfw_ax3 + (a + 3b + c)x2 + (b + c)x 3a : a, b, c R.
Math 250B
Group Work #11
SOLUTIONS
Fall 2013
Problem 1. Dene T : P2 (R) R via
T (p(x) = p(1).
(a): Show that T is a linear transformation.
SOLUTION: We must show that T respects addition and scalar multiplication:
T respects addition: Given polynomials p1
Math 250B
Group Work #10
SOLUTIONS
Fall 2013
Problem 1. Determine whether each set of vectors below is an orthogonal set. For
those that are, determine a corresponding orthonormal set of vectors.
(a): cfw_(2, 1, 1), (1, 1, 1), (0, 1, 1)
SOLUTION: We have
Math 250B
Group Work #13
SOLUTIONS
Fall 2013
Problem 1. Find the general solution to the dierential equation below,
but only determine the form of the particular solution yp (x).
y 6y + 25y = x2 .
SOLUTION: We begin by solving y 6y + 25y = 0, which has ch
Math 250B
Midterm I Information
Spring 2014
WHEN: Wednesday, February 26, in class
(no notes, books, calculators I will supply a table of integrals)
EXTRA OFFICE HOURS:
Sunday, February 23: 9:00 PM MIDNIGHT
Monday, February 24: 10:0012:00 NOON and 3:004:0
Math 250B
Group Work #14
SOLUTIONS
Fall 2013
Find the general solution to each dierential equation below:
Problem 1:
y + 4y + 4y =
ln x
xe2x
SOLUTION: The complementary function is
yc (x) = c1 e2x + c2 xe2x .
We set
y1 (x) = e2x
and
y2 (x) = xe2x .
We pro
Math 250B
Group Work #15
SOLUTIONS
Fall 2013
Problem 1. Discuss how you would solve each dierential equation below, or state that
no method we learned can be applied. Then go back and solve as many as you have
time for.
(a):
yx
dy
dy
= 3 2x .
dx
dx
(x > 0
Math 250B
Midterm II Information
Fall 2013
SOLUTIONS TO PRACTICE PROBLEMS
Problem 1. Determine whether each set S below forms a subspace of the
given vector space V . Show carefully that your answer is correct:
(a): V = P3 (R) and S = cfw_2x3 + bx + a : a
December 2013
Midterm III Review Session
Math 250B
SOLUTIONS
Name:
Problem 1. Suppose A is a square matrix such that
A3 2A2 + 4A + 5I = 0.
If is an eigenvalue of A, show that
3 22 + 4 + 5 = 0.
SOLUTION: There is some eigenvector v such that Av = v. Then
A
April 7, 2010
Midterm II
Math 250B
SOLUTIONS
Problem 1. Let
Name:
1 3 2
1
6 .
A = 3 10 4
2 5 6 1
(a): (9 points) Find a basis and the dimension for nullspace(A).
SOLUTION: We consider the augmented matrix and perform row operations:
1 3 2
1 0
1
3 2
1 0
1
Math 250B
Group Work #6
Fall 2013
Problem 1. Suppose A is an n n matrix such that A8 = 0n . Show that the matrix In + A2 is
invertible and
(In + A2 )1 = In A2 + A4 A6 .
Problem 2. For each item below, a vector space V is given. Decide (with justication) w
Math 250B
Group Work #8
Fall 2013
Goals: Practice spanning sets and linear dependence/independence.
Problem 1. Let V = M23 (R) and let W be the subset of V consisting of all 2 3
matrices whose rows and columns each add up to zero. Find a set of vectors th
Math 250B
Group Work #7
Fall 2013
Problem 1. Let V = M2 (R) and let S be the subset of V consisting of all 2 2 matrices whose
four entries add up to zero.
(a): Show that S is a subspace of V .
(b): Find a set of vectors that span S.
Problem 2. Let V = P2
Math 250B
Group Work #9
Fall 2013
Problem 1. Find a basis for the subspace S of P3 (R) consisting of
S = cfw_ax3 + (a + 3b + c)x2 + (b + c)x 3a : a, b, c R.
What is dim(S)?
Problem 2. Find a basis for the subspace S of M3 (R) consisting of all 3 3 matrice
Math 250B
Group Work #11
Fall 2013
Goals: Practice with linear transformations.
Problem 1. Dene T : P2 (R) R via
T (p(x) = p(1).
(a): Show that T is a linear transformation.
(b): If p(x) = ax2 + bx + c, then what is T (p(x) ?
(c): Use part (b) to nd Ker(T