a If A is a 4 4 matrix and the system Ax 2214 is inconsist

a) If A is a 4 × 4 matrix and the system A~x =[2;2;1;4 ] is inconsistent, what can you conclude about the dimension of N(A)? Explain your answer.

b) Let A be an m × n matrix. Explain why the row space and the column space of A have the same dimension.

c) Let A be an m × n matrix with a non-zero nullspace. Explain why the columns of A must be linearly dependent.

Solution

(a) If Ax =[2,2,1,4] is inconsistent, then det (A) has to be zero.

Otherwise, A will be an invertible matrix and Ax =[2,2,1,4] will have a (unique) solution. (and hence cannot be inconsistent).

As det (A)=0, N(A) = dimension of null space is at least one

(b) Row rank of A = Column rank of A\' (A\' denoting the transpose of A) and vice versa. As A and A\' have the same rank , the result follows.

(c) Dimension of the column space of A is the rank of A.

As A has non-zero null space , rank A < n. (dim V = N(A) + Rank (A))

It follows that the columns of A must be linearly dependent.