Question

If the null space of a 4 x 9 matrix is 5-dimensional, what is the dimension of the column space of the matrix? Explain your answer b) (3 points) If A is a 7 x 3 matrix, what is the smallest possible dimension of Nul A? Explain your answer

Answer 1

Answer : Dimension of column A is also be 4 whereas the two vector basis lie in R⁴.

The smallest possible dimension of Nul A would be zero.

Explanation:

Since we have given that

A is matrix of 4 x 9 .

so, Number of rows = 4

Number of columns = 9

Nul A = 5

It means that Rank of A would be 9 - 5 =4

So, rank A = 4

Thus, dimension of column A is also be 4 whereas the four vector basis lie in R⁴.

So, dim Col A = 4

If A is 7 x 3 matrix.

So, we know that

rank A + dim (null A) = 3

so, it is possible to have rank A = 3 so the dim col A should be 3

Then the smallest possible dimension of Nul A would be zero.