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.

 

[blue_raver22]

Yes, you can use the theorem you mentioned to prove that the column space of AB is contained in the column space of matrix A.

Firstly, we know that the column space of a matrix A is the span of its column vectors. This means that any vector in the column space of A can be written as a linear combination of the column vectors of A.

Now, let's consider the matrix multiplication AB. The resulting matrix AB will have column vectors that are linear combinations of the column vectors of A. This is because each column of AB is computed by multiplying A with a column vector of B. Therefore, the column space of AB will also be a span of the column vectors of A.

Next, we can use the theorem you mentioned to show that any vector in the column space of AB is also in the column space of A. Since the column space of AB is a span of the column vectors of A, any vector in this space can be written as a linear combination of the column vectors of A. This means that it is also a solution to the system of linear equations Ax=b, which implies that it is in the column space of A.

In conclusion, we have shown that the column space of AB is contained in the column space of A by using the fact that the column space of AB is a span of the column vectors of A and the theorem that states a system of linear equations has solutions if and only if the vector is in the column space of the coefficient matrix.