Vector space of the product of two matrices

In summary, the conversation discusses the attempt to prove that the product of a m x n matrix M with column space R^m and a n x o matrix N with column space R^n, MN, has column space R^m. The individual is unsure of where to start and suggests that any solution to M augmented with a vector can be tweaked to be a solution for MN. The conversation also clarifies that the column space should be R^m for matrix M and R^n for matrix N.
  • #1
redjoker
5
0
I'm trying to prove (as part of a larger proof) that the product of a m x n matrix M with column space R^m and a n x o matrix N with column space R^n, MN, has column space R^m. I'm not sure where to begin. What I'm thinking should be the right approach is to show that any solution to M augmented with a vector v = (a_1, ..., a_m) can be tweaked to be a solution for MN, though I haven't been able to get there. Any suggestions?
 
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  • #2
Did you really mean column space R^m for mxn matrix M and col space R^n for nxo matrix N ? Or was it supposed to be R^n for the former since matrix M has n columns?
 
  • #3
Yeah I meant column space. The assumption is that m,n,o form a non-decreasing sequence.
 

Related to Vector space of the product of two matrices

1. What is a vector space of the product of two matrices?

A vector space of the product of two matrices is the set of all possible linear combinations of the columns of the product matrix. This means that any vector in the product matrix can be expressed as a sum of the columns of the product matrix multiplied by scalar coefficients.

2. How is the vector space of the product of two matrices different from the original vector spaces?

The vector space of the product of two matrices is a new vector space that is created by combining the individual vector spaces of the two original matrices. It may have different dimensions and basis vectors than the original vector spaces.

3. What is the significance of the vector space of the product of two matrices in linear algebra?

The vector space of the product of two matrices is important in linear algebra because it allows for the representation of complex transformations in a concise and efficient manner. It also allows for the analysis and manipulation of these transformations using matrix operations.

4. Can the vector space of the product of two matrices be infinite-dimensional?

Yes, the vector space of the product of two matrices can be infinite-dimensional in some cases. This can occur when the original matrices have infinite-dimensional vector spaces or when the product matrix has an infinite number of columns.

5. How does the vector space of the product of two matrices relate to matrix multiplication?

The vector space of the product of two matrices is closely related to matrix multiplication. In fact, the columns of the product matrix are formed by multiplying the original matrices and then taking linear combinations of the resulting columns. This means that the vector space of the product of two matrices is a fundamental concept in understanding and working with matrix multiplication.

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