Question

Linear Algebra: Let C = AB. If the column vectors of B are linearly dependent, are the column vectors of C linearly dependent too?

Answer 1

is A another matrix?

Answer 2

yes.

Answer 3

of course it is. Since the rank of product of matrices is at most the rank of the min of the ranks. So, C would have a rank of minimum rank between A and B. Since B has rank less than it's number of columns, the rank of C has to be less than that number. Suppose the dimensions are
\[m \times n = m \times p . p \times n\]
since rank of B is less than n, rank of C has to be less than n, and hence, yes, C's columns are linearly dependent.

Answer 4

of course it is. Since the rank of product of matrices is at most the rank of the min of the ranks. So, C would have a rank of minimum rank between A and B. Since B has rank less than it's number of columns, the rank of C has to be less than that number. Suppose the dimensions are
\[m \times n = m \times p . p \times n\]
since rank of B is less than n, rank of C has to be less than n, and hence, yes, C's columns are linearly dependent.