Relationship of an invertible matrix in spanning set and linear independence

[ichigo444]

What could we say if a matrix is invertible? Could we say that it can span and is linearly independent?

 

[HallsofIvy]

I have no idea what you are talking about! "Span" and "linearly independent" are properties of sets of vectors, not matrices. Are you referring to the columns or rows as a vectors?

If so, then, yes, a matrix is invertible if and only if its columns (equivalently, rows) thought of as vectors in Rⁿ are independent.

But I still don't know what you mean by "can span". Can span what? Any set of vector spans something. It is true that if an n by n matrix is invertible then its columns (equivalently, rows) thought of as vectors in Rⁿ span Rⁿ.