Math 284
Cuyamaca College
Name:________________________
Instructor: Dan Curtis
Practice Exam 2
1) True or False. Justify your answers. (Any matrix listed is not assumed to be
square or invertible unless stated.)
a) If AB
AC and A 0, then B C.
b) If D is n n and the equation Dx b has no solution for some b
is not invertible.
c) Let T :
T (e 2 )
2
2
n
, then D
is the linear transformation such that T (e1 ) (3, 4) and
( 2, 7) , then T is both one-to-one and onto.
d) Let A be a 3 5 matrix. Then the columns of A could be linearly independent,
but they can’t span 3 .
2) a) Find the standard matrix, A, for the linear transformation given by:
2 x1 3x2
T:
2
3
, T ( x)
x2
x1 3x2
b) Determine whether T is one-to-one, onto or both. Justify your answer.
3) Determine whether the given transformation is linear. Justify your conclusion.
T:
2
2
,T
x1
5 x1 2 x2
x2
3x2
4) For problem 4, let A
1
2
2 3
,B
2
3
3
2
2
1
1 3 5
, and C
indicated calculation is not possible, indicate why.
a) Find AB.
b) Find BA
c) Find C-1.
d) Find det(C )
e) Find BT
2 4 6 . If the
1 7 8
5) Let T :
2
2
be the transformation that reflects a vector across the x1 axis
and then rotates it counterclockwise by an angle of
for the transformation.
. Find the standard matrix A
6) Suppose A and B are row equivalent n n matrices, and the following series of
row operations transforms A into B.
R1
2 R1
3R1
1
4
R3
R2
R3
R2
2 R2
R3
6 R3
If B
1 3 5
0 4 6 find det( A).
0 0 8
7) Suppose D, E and F are invertible nxn matrices and I is the nxn identity matrix.
Solve for E.
D 1ED 1 F I
8) Let T :
A
2
2
3
2
be the transformation T (x)
1
2
Ax, where
.
Show on the graph the result of applying the transformation to the image below.