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

In summary, To prove that the column space of AB is contained in the column space of matrix A, you can use the theorem "a system of linear equations Ax=b has solutions if and only if b is in the column space of A". This can be applied by showing that if y=(AB)x, then y=A(Bx), which means that y is also in the column space of A. This is a standard method for proving subspaces.
  • #1
mitch_1211
99
1
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?
 
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  • #2
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.
 
Last edited:
  • #3
td21 said:
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 */
 
  • #4
mitch_1211 said:
Sorry I'm not sure what you mean with the notation /* and */

sorry now changed.
 
  • #5


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.
 

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

1. What does it mean for the column space of AB to be contained in the column space of matrix A?

It means that every column vector in the column space of AB is also in the column space of matrix A. In other words, the column space of AB is a subset of the column space of matrix A.

2. How can you prove that the column space of AB is contained in the column space of matrix A?

One way to prove this is by showing that every column vector in the column space of AB can be written as a linear combination of the columns of matrix A. Another way is by showing that the column space of AB is a subspace of the column space of matrix A.

3. Why is it important to prove that the column space of AB is contained in the column space of matrix A?

This proof is important because it demonstrates that the columns of AB are linear combinations of the columns of matrix A, which shows a relationship between the two matrices. It also helps us understand the properties and behavior of matrix multiplication.

4. Can the column space of AB be equal to the column space of matrix A?

Yes, it is possible for the column space of AB to be equal to the column space of matrix A. This happens when the columns of AB are linear combinations of the columns of matrix A, and there are no additional columns in the column space of AB.

5. Are there any specific conditions that must be met for the column space of AB to be contained in the column space of matrix A?

Yes, for the column space of AB to be contained in the column space of matrix A, the number of rows in matrix B must be equal to the number of columns in matrix A. Additionally, the columns of AB must be linear combinations of the columns of matrix A.

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