- #1

- 3

- 0

## Main Question or Discussion Point

Q1:

Let A =

1 3 0

0 -4 4

3 4 5

a) Find Cartesian equations for Col(A) and Null(A).

b) Are the columns of A linearly independent? Give full reasons for your answer (based on

the definition of linear independence).

Q2: Prove that in any vector space k × 0 = 0, where k is any real number and 0 is the zero vector.

Q3:

The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.

The set of diagonal matrices D =

[(a 0

0 b) | a, b 2 E R is a subset of M2,2

a) Write down two particular matrices which belong to D, and two particular matrices which

belong to M2,2 but not to D.

b) Prove that D is a subspace of M2,2

Let A =

1 3 0

0 -4 4

3 4 5

a) Find Cartesian equations for Col(A) and Null(A).

b) Are the columns of A linearly independent? Give full reasons for your answer (based on

the definition of linear independence).

Q2: Prove that in any vector space k × 0 = 0, where k is any real number and 0 is the zero vector.

Q3:

The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.

The set of diagonal matrices D =

[(a 0

0 b) | a, b 2 E R is a subset of M2,2

a) Write down two particular matrices which belong to D, and two particular matrices which

belong to M2,2 but not to D.

b) Prove that D is a subspace of M2,2