Let be a field and let be a square matrix of dimension over . Prove that the set is linearly independent if and only if .
Let be the reduced row echelon form of , and let be invertible such that .
Suppose the columns of are linearly independent. Now has column rank . In particular, . Now ; so .
We prove the converse contrapositively. Suppose the columns of are linearly dependent; then the column rank of is strictly less than , so that has a row of all zeros. Using the cofactor expansion formula, . Since is invertible, its determinant is nonzero; thus . Thus if , then the columns of are linearly independent.