Suppose a 4x7 matrixA has four pivot columns Is ColA R4 Is

Suppose a 4x7 matrixA has four pivot columns. Is ColA = R⁴? Is NulA = R³? Explain.

Solution

If \"A\" has 4 pivot columns, then the basis for Col A is a set of 4 Linearly Independent vectors in R⁴. Thus, that basis must span R⁴. A basis for Nul A does feature a set of 3 Linearly Independent vectors that span a subspace of dimension 3, but this set is in R⁷, not R³.