Show that matrix AB has its row and column vectors in A and B

In summary, the question asks to show that the row vectors of AB are in the row space of B and the column vectors of AB are in the column space of A. This can be proven using the definition of matrix multiplication, where the product of a row of A and a column of B is equal to the corresponding entry in the resulting matrix AB. This means that the row vectors of AB can be expressed as a linear combination of the rows of B, and the column vectors of AB can be expressed as a linear combination of the columns of A.
  • #1
jheld
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Homework Statement


Given matrices A and B, show that the row vectors of AB are in the row space of B an the column vectors of AB are in the column space of A


Homework Equations


Just matrix multiplication, reduced row echelon form, and leading one's for row and columns


The Attempt at a Solution


I am unsure if I need to find the two separate matrices, or if this is just a general problem.
I understand what they are asking, but I am unsure of how to find the right matrices, and what it means to be "in" the row and column vectors. I think it means that they are in the correct rows and columns from the respective single matrices A and B.
 
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  • #2
It means that a row of AB is a linear combination of rows of B and a column of AB is a linear combination of columns of A. All you need is the definition of matrix multiplication, A_{ij}*B_{jk}=(AB)_{ik}.
 

1. How do you show that matrix AB has its row and column vectors in A and B?

To show that matrix AB has its row and column vectors in A and B, you can use the fact that the rows of AB are linear combinations of the rows of A using the columns of B. Similarly, the columns of AB are linear combinations of the columns of B using the rows of A. This demonstrates that the row and column vectors of AB are contained within A and B respectively.

2. Can you provide an example to illustrate this concept?

Yes, for example, if we have matrix A with rows [1 2] and [3 4] and matrix B with columns [5 6] and [7 8], then the resulting matrix AB would have rows [19 22] and [43 50], which are linear combinations of the rows of A using the columns of B. Similarly, the columns of AB would be [23 34] and [31 46], which are linear combinations of the columns of B using the rows of A.

3. What is the significance of showing that matrix AB has its row and column vectors in A and B?

This concept is important because it helps us understand the relationship between matrices and how operations on them affect their row and column vectors. It also helps us in finding the inverse of a matrix, as the inverse of a matrix can be found using the row and column vectors of the original matrix.

4. Are there any other methods to show that matrix AB has its row and column vectors in A and B?

Yes, another method is to use the properties of matrix multiplication. For example, we know that the rows of AB are linear combinations of the rows of A using the columns of B. We can also say that the rows of AB are linear combinations of the columns of B using the rows of A. These properties can be used to prove that the row and column vectors of AB are contained within A and B respectively.

5. How does this concept relate to the concept of matrix multiplication?

This concept is directly related to the concept of matrix multiplication, as it helps us understand the properties and effects of multiplying two matrices. It also shows that the result of matrix multiplication has its row and column vectors contained within the original matrices, which can be useful in various applications of matrices in fields such as physics, engineering, and computer science.

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