Relationship of an invertible matrix in spanning set and linear independence

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

HallsofIvy

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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.
 

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