Answer to Question #101032 in Linear Algebra for Derty

Question

1. Use the information in the table to:

(i) find the dimensions of the row space of A, column space of A, null space of A, and null space of A Superscript T ;

(ii) determine whether or not the linear system Ax = b is consistent;

(iii) find the number of parameters in the general solution of each system in (ii) that is consistent.

Size of A= 9 × 11

Rank (A) =2

Rank ([A | b])= 3

i) The dimension of the row space of A is?

The dimension of the column space of A is?

The dimension of the null space of A is?

The dimension of the null space of AT is

(ii) The linear system _____ consistent.

(iii) The number of parameters in its general solution is

Answer 1

• The column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A. And column rank = row rank = rank (A)

i) The dimension of the row space of A is 2

The dimension of the column space of A is 2

• Theorem: If A is m  x n, then rank (A) + nullity (A) = n.

Also it is known that rank(A)=rank(AT)

nullity(A)=11-2=9

nullity(AT )=9-2=7

i) The dimension of the null space of A is 9

The dimension of the null space of AT is 7

• rank (A) ≠ rank ([A|b]) $\Rightarrow$ the linear system isn’t consistent

(ii) The linear system isn’t consistent.

(iii) The number of parameters in its general solution is 0