Suppose a 5x6 matrixA has four pivot columns What is dim Nul

Suppose a 5x6 matrixA has four pivot columns. What is dim NulA? Is ColA = R⁴? Why or why not?

Solution

We know that rank(A) + dim(Nul(A)) = 6, and rank(A) is the number of pivot columns of A. If A has 4 pivots, then it must be the case that dim(Nul(A)) = 2.Furthermore, if dim(Col(A)) = 4, we know that Col(A) has a basis of four elements. It does turn out that Col(A) is isomorphic to R⁴ . It is not equal to R⁴, however, because it is a subspace of R⁵ (the matrix has 5 rows).