Let A and B be matrices such that the matrix product AB is d

Let A and B be matrices such that the matrix product AB is defined. Show that if the columns of B are linearly dependent, then so are the columns of AB. It is not enough to give an example for a specific matrix B.

Solution

The columns of a matrix B are linearly dependent if and only if Bx = 0 has a nontrivial solution.... (i)
If x is a solution of Bx = 0, it is a solution to ABx = 0:

ABx => A (Bx) => A (0) => 0.

Thus ABx = 0

By statement (i), the columns of AB are linearly dependent.

Here is essentially the same proof as above:

Let B = [b1 · · · bn]

as before,The columns of B are linearly dependent if and only if there
exist 1, . . . , n R (at least one nonzero) such that

1b1 + 2b2 + · · · + nbn = 0....... (ii)

After matrix multiplication,
AB = [Ab1 Ab2 · · · Abn]

i.e.,

the columns of AB are given by A multiplied by the columns of B.

Thus the columns of AB are linearly dependent:

ABx => 1Ab1 + 2Ab2 + · · · + nAbn

=> A (1b1) + A (2b2) + · · · + A (nbn)

=> A (1b1 + 2b2 + · · · + nbn)

By statement (ii)
ABx = A (0) = 0

Thus ABx = 0 and this proves that the columns of AB are linearly dependent.