Let Aequals[Bold v 1 Bold v 2 Bold v 3 Bold v 4]. Note that A is a ▼ 1 times 4 4 times 4 matrix and its columns span ▼ set of real numbers R Superscript 4 Baseline . set of real numbers R . Thus, by the ▼ definition of linear independence, Basis Theorem, Spanning Set Theorem, definition of a basis, Invertible Matrix Theorem, Rank Theorem, the columns ▼ are linearly dependent. are pivot columns. span set of real numbers R cubed . are linearly independent. Therefore, the columns of A are a basis for set of real numbers R Superscript 4 because of the