Final answer:
The columns of a matrix A span when A is invertible because an invertible matrix has a full rank, which means that its columns are linearly independent and span the entire vector space.
Step-by-step explanation:
The columns of a matrix A span when A is invertible because an invertible matrix has a full rank, which means that its columns are linearly independent and span the entire vector space.
When A is invertible, it means that for every vector b in the vector space, the equation Ax = b has a unique solution. This implies that any vector in the vector space can be expressed as a linear combination of the columns of A, thus spanning the vector space.
For example, consider a 3x3 invertible matrix A. The columns of A, denoted as A1, A2, and A3, form a basis for the vector space. This means that any vector in the vector space can be represented as a linear combination of A1, A2, and A3.