Why are base-change matrices always invertible?

Question

I noticed something strange in my linear algebra lecture notes: the notes state that the base-change matrix that has m rows and n columns will always be invertible.. but how can that be? If m doesn't equal n how can the matrix be invertible?

Answer 1

They are bases of the same vector space $V$, and the dimension of a vector space is equal to the cardinality of any basis of that vector space. It follows that $|B| = |C|$ giving $m = n$.