If the null space of an 8 times 7 matrix is 5dimensional wha

If the null space of an 8 times 7 matrix is 5-dimensional, what is the dimension of the column space? If a 6 times 8 matrix has four pivot columns, what is the dimension of its null space? Is the column space R^4? Why or why not? If A is a 3 times 7 matrix, what is the smallest possible dimension of Null A? Is it possible that the transformation x rightarrow x is one-to-one?

Solution

2) If null space has dimension 5 then columns space = 8 - 5 = 3

3) If 6 x 8 has 4 pivots then column space has dimension 4

So, null space has dimension 2

4) 3x7 matrix. Smallest possible null space = 0