Prove the column space of AB is contained in the column space of matrix A

[mitch_1211]

lets assume the matrix multiplication AB exists, how would i prove that the column space of AB is contained in the column space of matrix A?

i know there is a theorem that says something like: "a system of linear equations Ax=b has solutions if and only if b is in the column space of A"

Am i to use something similar to this here?

 

[td21]

if y=(AB)x [i.e y in the column space of AB ]
then y=A(Bx) [i.e y in the column space of A ]

This is the typical way of the subspace proof.

 

[mitch_1211]

if y=(AB)x /* y in the column space of AB */
then y=A(Bx) /* y in the column space of A */

This is the typical way of the subspace proof.

Sorry I'm not sure what you mean with the notation /* and */

 

[td21]

Sorry I'm not sure what you mean with the notation /* and */

sorry now changed.