## Description

1. Let T : R

n → R

m be a linear transformation.

(a) Show that the null space of T is a subspace of R

n.

(b) Show that the range of T is a subspace of R

m.

2. (a) For a 4 × 3 matrix M, must the column space of M be identical to the column space of

rref(M)?

(b) For a 3 × 3 matrix N with rank(N) = 3, must the column space of N be identical to the

column space of rref(N)? Can the assumption that rank(N) = 3 be dropped?

3. For a linear transformation L : R

3 → R

3

, we have the following information:

L

1

1

0

=

2

1

1

L

−1

1

0

=

−3

3

3

L

0

0

1

=

0

0

0

.

(a) Write down a matrix for L.

(b) Describe the range of L as a point, line, plane, or hyperplane and give a basis for the

range of L.

(c) Describe the null space of L as a point, line, plane, or hyperplane and give a basis for the

null space of L.

1