Show colsp(AB) is contained in colsp(A)

  • Thread starter Robb
  • Start date
In summary, the column space of AB is contained in the column space of A because the column space of AB consists of all linear combinations of the columns of A, and matrix multiplication is associative. Therefore, any vector in the column space of AB can also be written as a linear combination of the columns of A.
  • #1
Robb
225
8

Homework Statement


Let A and B be matrices for which the product AB is defined. Show that the column space of AB is contained in the column space of A.

Homework Equations


perform one elementary row operation to matrix A to obtain matrix B.

The Attempt at a Solution



1 2
3 4 = A

2 4
6 8 = 2A = B

14 20
30 44 = AB

Obviously the colsp(A) = colsp(B) given that B is a linear combination of A, but I don't see how colsp(AB) = colsp(A) and is therefore contained in A. Please advise.
 
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  • #2
To make this easy: Let's assume both matrices are ##n## x ##n##

Suppose I block ##\mathbf B## by column. I can now do matrix vector multiplication one column at a time in ##\mathbf B##.

##\mathbf {AB} = \mathbf A \bigg[\begin{array}{c|c|c|c|c}
\mathbf b_1 & \mathbf b_2 &\cdots & \mathbf b_{n-1} & \mathbf b_{n}
\end{array}\bigg] = \bigg[\begin{array}{c|c|c|c|c} \mathbf A \mathbf b_1 & \mathbf A\mathbf b_2 &\cdots & \mathbf A\mathbf b_{n-1} & \mathbf A\mathbf b_{n}
\end{array}\bigg]##

now let's look at the ##kth## column of the Right Hand side. Block ##\mathbf A## by column and look at ##\mathbf A \mathbf b_k## -- what does that tell you?
Robb said:
Obviously the colsp(A) = colsp(B) given that B is a linear combination of A

I didn't find this obvious or correct. For example: ##\mathbf A## could be singular and ##\mathbf B## could be non-singular.

The relevant equation, in my view, is matrix vector and matrix matrix multiplication. I'm not sure what elementary row operations have to do with this. It seems like something is missing from the problem statement.
- - - -
edit: If you're going to use a blocked-multiplication argument and respond in the forums, you must use Tex/ LaTeX. The forum sticky is here:

https://www.physicsforums.com/help/latexhelp/

I also like the GUI approach here:
https://www.codecogs.com/latex/eqneditor.php
 
  • #3
1. follows from the definition of matrix multiplication. I.e. when you multiply a matrix A by a column vector v the result Av is the column vector which is the linear combination of th columns of A, using the entries of v as coefficients. Hence whenever you multiply a matrix A by a sequence of columns, the columna of B, the result AB is a matrix whose columns are all linear combinations of the columns of A. In particular every linear combination of the columns of AB is also a linear combination of the columns of A. QED.If you say this more conceptually, the column space of A consists of all column vectors which can be written as Av for some column vector v. Similarly a vector in the column space of AB is one that can be written as (AB)w for some w. But, since matrix multiplication is associative, any such vector (AB)w can also be wriiten as A(Bw), so if we write v for Bw, the vector (AB)w in the column space of AB, also has form Av, hence belongs also to the column space of A.
 
  • #4
mathwonk said:
1. follows from the definition of matrix multiplication. I.e. when you multiply a matrix A by a column vector v the result Av is the column vector which is the linear combination of th columns of A, using the entries of v as coefficients. Hence whenever you multiply a matrix A by a sequence of columns, the columna of B, the result AB is a matrix whose columns are all linear combinations of the columns of A. In particular every linear combination of the columns of AB is also a linear combination of the columns of A. QED.If you say this more conceptually, the column space of A consists of all column vectors which can be written as Av for some column vector v. Similarly a vector in the column space of AB is one that can be written as (AB)w for some w. But, since matrix multiplication is associative, any such vector (AB)w can also be wriiten as A(Bw), so if we write v for Bw, the vector (AB)w in the column space of AB, also has form Av, hence belongs also to the column space of A.
Thank you!
 
  • #5
StoneTemplePython said:
To make this easy: Let's assume both matrices are ##n## x ##n##

Suppose I block ##\mathbf B## by column. I can now do matrix vector multiplication one column at a time in ##\mathbf B##.

##\mathbf {AB} = \mathbf A \bigg[\begin{array}{c|c|c|c|c}
\mathbf b_1 & \mathbf b_2 &\cdots & \mathbf b_{n-1} & \mathbf b_{n}
\end{array}\bigg] = \bigg[\begin{array}{c|c|c|c|c} \mathbf A \mathbf b_1 & \mathbf A\mathbf b_2 &\cdots & \mathbf A\mathbf b_{n-1} & \mathbf A\mathbf b_{n}
\end{array}\bigg]##

now let's look at the ##kth## column of the Right Hand side. Block ##\mathbf A## by column and look at ##\mathbf A \mathbf b_k## -- what does that tell you?

I didn't find this obvious or correct. For example: ##\mathbf A## could be singular and ##\mathbf B## could be non-singular.

The relevant equation, in my view, is matrix vector and matrix matrix multiplication. I'm not sure what elementary row operations have to do with this. It seems like something is missing from the problem statement.
- - - -
edit: If you're going to use a blocked-multiplication argument and respond in the forums, you must use Tex/ LaTeX. The forum sticky is here:

https://www.physicsforums.com/help/latexhelp/

I also like the GUI approach here:
https://www.codecogs.com/latex/eqneditor.php
Gracias!
 

1. What does "colsp(AB)" mean in the context of this statement?

"colsp(AB)" refers to the column space of the matrix product AB. This means all possible linear combinations of the columns of AB.

2. How is the column space of AB related to the column space of A?

The column space of AB is a subset of the column space of A. This means that all the columns of AB are also columns of A, but A may have additional columns not present in AB.

3. Can you provide an example to illustrate this statement?

Yes, let's say we have matrix A with columns [1,2,3] and [4,5,6], and matrix B with columns [7,8] and [9,10]. The product AB would have columns [31,41] and [46,56]. The column space of AB would be all possible linear combinations of these two columns, while the column space of A would include these two columns plus any additional columns present in A.

4. How can we prove that "colsp(AB) is contained in colsp(A)"?

We can prove this by showing that any column in the column space of AB is also present in the column space of A. This can be done by using the definition of column space and showing that each column in AB is a linear combination of columns in A.

5. What implications does this statement have in linear algebra?

This statement shows that the column space of a matrix product is always a subset of the column space of the individual matrices involved. This is important in understanding the properties and relationships between matrices and their column spaces in linear algebra.

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