name: math 211 :: exam guide :: november 11, 2016

Name: Math 211 :: Exam Guide :: November 11, 2016

1. Answer the following questions

M1 =

.5 4.8

.1 8.4

�

8>>>><

>>>>:

What is the basis for the null space?

What is the basis for the column space?

M2 =

2

664

5 1 6

�2 2 0

1 3 4

3 6 9

3

775

8>>>><

>>>>:



M4 =

3 2 1

2 2 1

�

8>>>><

>>>>:



1


2. Dealings with Eigenvalues.

(a) Find the eigenvalues and eigenvectors of the following ma-

trices and describe the image of the unit square under the

linear transformation defined by each of these matrices.

(b) Plot the eigenvectors on the same axis as the image of the unit

square.

(c) Use the eigenvalues to compute the determinant of each matrix.

(d) Estimate A

100v and B

100if v =

.1

.9

�

A =

.6 .3

.4 .7

�

B =

2 4

0 0

�

2


3. Find the eigenvalues and eigenvectors of the following matrix.

2

40 2 �20

1 0 19

0 1 0

3

5

3


4. Determine which of the following transformations are linear.

If linear, find the standard matrix for the transformation. If non-linear,

construct an example to justify non-linearity.

Transformation Matrix

� �

T (x, y) = (3x, x� y, x+ y)

T (x, y, z) =

px

2+ y

2+ z

2

4


5. Suppose that T : R3 ! R3is an invertible linear transformation and

T (1, 0, 0) = (1, 2, 3), T (0, 1, 0) = (1,�1, 1), T (7, 8, 9) = (0, 0, 1)

Compute T (4, 7,�1)

Find the standard matrix for T .

5

name: math 211 :: exam guide :: november 11, 2016

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