Math 54 quiz solutions
October 12, 2009
1 Dene the null space of a matrix (3 points).
If A is an n m matrix, the null space of A is the set of vectors x in Rm
such that Ax = 0.
2 Dene what it means for a set of vectors to be linearly dependent (3 points).
Math 54 worksheet, September 21, 2009
1. Find the inverse of
1
A = 3
2
2
4 ,
4
0
1
3
using both row reduction and the adjugate. Check that you get the same
answer.
A1
8
= 10
3
4
3
2
7
2
1
1
1
2
2. Use the adjugate to nd a formula for the inverse of a 2 2
Math 54 worksheet, September 30, 2009
1. Write the solutions to the following system of equations in parametric
vector form:
3x1 + 2x2 + 2x3 = 7
2x1 2x2 + 8x3 = 8
x1 + 4x2 6x3 = 1
The solutions are:
x1
3
2
x2 = 1 + t 2
x3
0
1
2. Find bases for the row
Math 54 quiz solutions
October 30, 2009
1 Dene eigenspace (2.5 points).
For a matrix A and an eigenvalue of A, the -eigenspace is the set of all
vectors x such that Ax = x.
2 Dene what it means for a matrix to be diagonalizable (2.5 points).
A matrix A is
Math 54 quiz solutions
December 2, 2009
1 Find the Fourier cosine series expansion of f (x) = 3 + x dened on 0 < x <
(7 points).
For n > 0, we compute an using the formula:
an =
2
(3 + x) cos nx dx
0
Use integration by parts with u = 3 + x, dv = cos nx,
Math 54 quiz solutions
November 23, 2009
1 Find a general solution to the system: (5 points)
x (t) =
1
3
2
1
x(t) +
.
2
1
The characteristic equation of the matrix is:
1
3
2
= 2 3 + 2 6 = 2 3 4 = ( 4)( + 1),
2
so the eigenvalues are 4 and 1. For = 4,
3
3
Math 54 quiz solutions
November 18, 2009
1 Find a general solution (6 points) for the system
x (t) = Ax(t)
A=
13
.
12 1
Find the solution with initial conditions x(0) =
2
(4 points).
0
The characteristic polynomial of A is:
1
12
3
= 1 2 + 2 36 = 2 2 35 =
Math 54 quiz solutions
November 4, 2009
1 Find the general solution to the dierential equation (5 points)
y + y 2y = e2t
The auxillary equation of the homogeneous dierential equation is r2 + r 2 =
(r + 2)(r 1) so the roots are r = 2 and r = 1. Thus, the g
Math 54 quiz solutions
October 28, 2009
1 Find the equation y = 0 + 1 x of the least-squares line that best ts the data
points: (1, 0), (2, 1), (4, 2), (5, 3). (5 points)
We have the matrices:
1
1
X=
1
1
0
1
.
and y =
2
3
1
2
4
5
The normal equations ar
Math 54 quiz solutions
October 21, 2009
1 Find an invertible matrix P and a matrix C such that
5
1
5
= P CP 1
1
where C =
a
b
b
.
a
(4 points)
The characteristic equation is
5
1
5
= 5 6 + 2 + 5 = 2 6 + 10.
1
By the quadratic formula, its roots are:
6 36 4
Math 54 quiz solutions
October 14, 2009
1 Find matrices P and D, with D diagonal, such that: (4 points)
2
4
3
= P DP 1 .
1
The characteristic equation of the matrix is
2
4
3
= 2 3 + 2 12 = 2 3 10 = ( 5)( + 2).
1
(The factorization was found by looking for
Math 54 quiz solutions
September 25, 2009
1 Use Cramers rule to nd the value of x3 in the solution to the equation:
3 2 5 x1
2
1 1 3 x2 = 1
6
1 6 x 3
3
Call the matrix on the left hand side A. We compute the determinant of A
by expanding by minors along
Math 54 quiz solutions
September 23, 2009
1 Find the determinant of
h
A= 3
4
2
1 .
1
3
6
5
(4 points). For what values of h is A invertible (2 points)?
We expand by minors along the rst row:
det A = h
6
5
3
1
3
4
1
3
1
+2
4
1
6
5
= h(6 5) 3(3 + 4) + 2(15
Math 54 quiz solutions
1 Here is a matrix and
1
1
A=
2
4
September 16, 2009
its echelon form:
2
1
0
1
1
954
6 5 3 0
6 1 2 0
0
9 1 9
2
1
0
0
9
3
0
0
54
0 7
1 2
00
Find bases for Col(A) and Nul(A) and state the dimensions of these subspaces.
(6 points)
A ba
Math 54 quiz solutions
September 9, 2009
1. Describe the solutions to the following in parametric vector form: (3 points)
x
3 1 4 0 1
0
2 1 2 5 x2 = 0
x3
1113
0
x4
We use elementary row operations to row reduce the matrix:
3140
1113
1113
2 1 2 5 2 1 2 5
Math 54 worksheet, September 14, 2009
1. Let T be the linear transformation from R3 to R3 which consists of rst
rotating 30 degrees about the z axis and then rotating 90 degrees about the
x axis. (I havent specied the directions of the rotations. Use whic