The column space of a 5 times 6 matrix is spanned by the vec

The column space of a 5 times 6 matrix is spanned by the vectors (1, 0, 0, 0, 0), (0, 0, 1, 0, 0), and (2, 0, 3, 0, 0). Find the dimension of the solution space of the matrix. 3 4 6 2 5

Solution

Given that column space = { ( 1,0,0,0,0), ( 0,0,1,0,0),(2,0,3,0,0) }

But 2 ( 1,0,0,0,0) + 3( 0,0,1,0,0) = (2,0,3,0,0)

So, ( 1,0,0,0,0), ( 0,0,1,0,0),(2,0,3,0,0) are linearly dependent.

Hence basis of column space = { ( 1,0,0,0,0), ( 0,0,1,0,0) }

So, dimension of column space = 2 = rank A

Sinc, we know that

dimension of solution space of A + rank A = number of columns of A

dimension of solution space = 6 - 2 = 4

Hwncw, (B) is correct answer.