MTH 201: Linear Algebra

Problem Sheet 6

September 10, 2014

1. Find bases of the image and kernel of the matrix

A =

1 2 0 1 21 2 0 2 31 2 0 3 41 2 0 4 5

2. Determine whether the following vectors form a basis of R4.

v1 =

1111

v2 =

1111

v3 =

1248

and v4 =

1248

3. Give an example of a 4 5 matrix A with dim(kerA) = 3.4. (a) Consider a linear transformation T from R5 to R3. What are the

possible values of dim(kerT ) = 3? Explain.

(b) Consider a linear transformation T from R4 to R7. What are thepossible values of dim(kerT ) = 3? Explain.

5. Find the matrix B of the linear transformation T (~x) = A~x with respectto the basis B=(~v1, ..., ~vm).

(a) A =

[0 12 3

]; ~v1 =

[12

], ~v2=

[11

].

(b) A =

4 2 42 1 24 2 4

; ~v1 = 212

, ~v2=02

1

, ~v3=10

1

.6. Let T be the linear transformation from R3 to R3 that reflects about the

line in R3 spanned by

123

. Find a basis B of Rn such that the B-matrixB of the given linear transformation is diagonal.

7. Consider a linear transformation T from R2 to R2. We are told thatthe matrix of T with respect to the basis

[01

],

[10

]is

[a bc d

]. Find the

standard matrix of T in terms of a, b, c and d.

1