Question

If the null space of A4×5 is exactly a two-dimensional plane, answer the following:

i. What is the dimension of the null space of A?
ii. What is the rank of A?
iii. How many linearly independent columns does A have?
iv. Does the system Ax = b sometimes have no solution?
v. Does Ax = b sometimes have a unique solution?
vi. Does Ax = b sometimes have [infinity]-many solutions

Answer 1

I.

A is a 4 x 5 matrix => A: U -> V, dim U = 5, dim V = 4

Null space is exactly two dimensional plane

dim null (A) = 2

II.

Rank A = dim U - dim Null A = 5 - 2 = 3

III.

Number of linearly Independent columns of A is the rank of A = 3

IV.

Yes, The system Ax = b has no solution sometimes as range of A \\eq V

V.

Yes,Sometimes Ax = b has a unique solution

VI.

Yes, sometimes Ax = b has infinitely many solutions