NAME_____________________________________________________________________________________________________
MAYNARD
MATH 285 TEST 3
LINEAR TRANSFORMATIONS, EIGENVALUES & EIGENVECTORS
SAMPLE
1)
Use T(v1, v2, v3) = (4v2 − v1, 4v1 + 5v2) to find
(a) the image of v for v = (2, −3, −1) and
(b) the preimage of w for w = (3, 9).
2)
Is T: P2  P2, T(a0 + a1x + a2x2) = (a0 + a1 + a2) + (a1 + a2)x + a2x2
a linear transformation?
3)
The linear transformation T: Rn Rm is defined by T(v) = Av,
for
 1 2
A   2 4 .
 2 2
Find
a) the image of (2, 4) and
b) the preimage of (−1, 2, 2) and then
c) explain why the vector (1, 1, 1) has no preimage under
this transformation.
4)
Let Dx be the linear transformation from C'[a, b] into C[a, b]
defined by Dx (f) = df/dx . Decide whether each of the
following statements is true or false.


 

 
(a ) D x e x  2 x  D x e x  2D x x 

2
2
(b) D x x 2  ln x  D x x 2  D x ln x 
(c) D x sin 2x   2D x sin x 
5)
Let T be the function from R2 into R2 such that
T(u) = projv u, where v = (1, 1). Find T(x, y).
6)
Find the kernel of T:P3  R, T(a0 + a1x + a2x2 + a3x3) = a0.
7)
The linear transformation
for
2 1
1
3
1
2
A
 4  3  1

1  2 1
T is given by T(v) = Av,
4
 1
 3

1
(a) Find a basis for the kernel of T, and
(b) Find a basis for the range of T.
8)
The linear transformation
for
2 2  3 1
1 1 1 1
A
3 3  5 0

6 6  2 4
(a) ker (T)
(b) nullity (T)
(c) range (T)
(d) rank (T)
T is given by T(v) = Av,
13 
 1 . Find
14 

16 
1
9)
Let T: P2  R be given by
Tp   x  px dx.

0
What is the kernel of T?
10)
Let T: R3  R3 be the linear transformation that
projects u onto v = (2, −1, 1). Find
(a) the rank and nullity of T, and
(b) a basis for the kernel of T.
11)
Use the standard matrix for the linear transformation
T(x, y, z) = (13x − 9y + 4z, 6x + 5y − 3z) to find
the image of the vector v = (1, −2, 1).
12)
a) Find the standard matrix A for the linear
transformation T and
b) use A to find the image of the vector v = (1, 4)
and
c) sketch v and its image, where T is defined as
T: R2  R2 is the reflection through the vector w = (3, 1)
in R2. [The reflection of a vector v through a vector w
is given by T(v) = 2 projw v − v. See diagram.]
v
v − projw v
w
projw v
projw v − v
−
projw v + (projw v − v) = 2 projw v − v
13)
Find T(v)
(a) by using the standard matrix and
(b) by using the matrix relative to B and B'., where
T: R3  R2 , T(x, y, z) = (x − y, y − z), v = (1, 2, −3),
B = {(1, 1, 1), (1, 1, 0), (0, 1, 1)}, B' = {(1, 2), (1, 1)}.
14)
Let T: P2  P3 be given by T(p) = xp. Find the matrix
of T relative to the bases
B = {1, x, x2} and B' = {1, x, x2, x3}.
15)
Let B = {1, x, ex, xex} be a basis of a subspace W
of the space of continuous functions, and let Dx be the
differential operator on W. Find the matrix for D x
relative to the basis B.
16)
Verify that i is an eigenvalue of A and that
xi is a corresponding eigenvector.
  2 2  3
A   2
1  6
  1  2 0 
1 = 5, x1 = (1, 2, −1)
2 = −3, x2 = (−2, 1, 0)
3 = −3, x3 = (3, 0, 1)
17)
Find
a) the characteristic equation and
b) the eigenvalues and corresponding
eigenvectors of the matrix
2 3
A

1 4 
18)
Find
a) the characteristic equation and
b) the eigenvalues and corresponding
eigenvectors of the matrix
 1 2  2
A   2 5  2
 6 6  3
19)
Let T: P2  P2 be given by
T(a0 + a1x + a2x2) = (−3a1 + 5a2) + (−4a0 + 4a1 − 10a2)x + 4a2x2.
Find the eigenvalues and corresponding eigenvectors
of T relative to the standard basis {1, x, x2}.
20)
Let T:M2,2  M2,2 be given by
 a b    a  c  d
bd 

T 

.

 c d    2a  2c  2d 2b  2d 
Find the eigenvalue s and eigenvecto rs of T
relative to the standard basis
1 0 0 1 0 0 0 0 
B  
, 
, 
, 
 .
0 0 0 0 1 0 0 1 
ANSWERS
1)
a) (−14, −7)
13)
b) (1, 1, t)
2)
Yes, it is linear.
3)
a) (10, 12, 4)
  4
1  
0 1
6
b) 
  5    7 
0

1

2

 1 
 B
  B'
 6(1, 2)  7(1, 1)  (1, 5)
b) (−1, 0)
c) (1, 1, 1) has no preimage because the system
1x + 2y = 1
−2x + 4y = 1
−2x + 2y = 1
14)
is inconsistent and therefore has
no solution.
4)
a) True
b) True
5)
1
6)
{a1x + a2x2 + a3x3: a1, a2, a3 are real}
7)
a) {(1, 1, 1, 0)}
b) {(1, 3, 4, 1), (2, 1, 3, 2), (4, 1, 3, 1)}
/2(x + y, x + y)
15)
a)
b)
c)
d)
{(2s − t, t, 4s, −5s, s): s and t are real}
2
{(7r, 7s, 7t, 8r + 20s + 2t): r, s and t are real}
3
9)
Span ({2−3x, 1 − 2x2}).
10)
a) Rank = 1, nullity = 2
b) {(1, 0, −2), (1, 2, 0)}
11)
 35 
 7 
 
a)  4
5
3

5
3 
5 
 4

5 
16 )
0 0
0 0
1 0

0 1
0
0
T : B, B  
0

0
1 0 0
0 0 0
0 1 1

0 0 1
  2 2  3  1 
1
2



1  6  2   5 2 

  1  2 0   1
 1
  2 2  3   2 
  2
2
  1   3 1 
1

6

 
 
  1  2 0   0 
 0 
  2 2  3  3
3
2



1  6 0  30

  1  2 0  1
1
17)
a) 2 − 6 + 5 = 0
b)  = 1, v1 = (3, −1);  = 5, v5 = (1, 1)
18)
a) 3 − 32 − 9 + 27 = 0
b)  = −3, v−3 = (1, 1, 3);
 = 3, v3 = (1, 0, −1) and v3 = (1, 1, 0)
19)
 = −2, p−2 = 3 + 2x
 = 4, p4 = −5 + 10x + 2x2
 = 6, p6 = −1 + 2x
b) (16/5, −13/5)
y
c)
0
1
T : B, B'  
0

0
c) False
8)
12)
1
1  1 0     1
a) 
 2    
0 1  1  3  5 
 
 1(1, 0)  5(0, 1)  (1, 5)
v = (1, 4)
w = (3, 1)
x
T(v) = (16/5, −13/5)
20 )
1 0
1 1 
  0, m 0  
and m 0  


1 0
0  1
 1 0
  3, m 3  

  2 0