Final answer:
The columns of an nxn matrix A span Rⁿ when the columns are linearly independent.
Step-by-step explanation:
The columns of an nxn matrix A span Rⁿ when the columns are linearly independent. This means that no column can be written as a linear combination of the other columns. When this condition is satisfied, the columns of the matrix A form a basis for Rⁿ, meaning that any vector in Rⁿ can be expressed as a linear combination of the columns of A.
For example, consider a 2x2 matrix A with columns [a b] and [c d]. If no multiple of one column can be written as the other column, then the columns are linearly independent and span R². Conversely, if one column can be written as a multiple of the other column, then the columns are linearly dependent and do not span R².