Are Matrix Multiplication Rules the Same for Composing Linear Transformations?

[1MileCrash]

So as I'm preparing for finals, I'm wondering:

The multiplication of two matrices is only defined under special circumstances regarding the dimensions of the matrices.

Doesn't that require that compositions of linear transformations are only defined in the same circumstances? I can't imagine not being able to not define a composition of linear transformations, can someone demonstrate this?

 

[economicsnerd]

If $f:V\to W$ and $g:X\to Y$ are linear transformations, it only makes sense to talk about the composition $g\circ f$ if $W=X$. In particular, if $f:\mathbb R^K \to \mathbb R^L$ and $g:\mathbb R^J \to \mathbb R^I$ are linear transformations, it only makes sense to talk about the composition $g\circ f$ if $\mathbb R^L=\mathbb R^J$, i.e. if $L=J.$

Phrasing the last point a different way now: If $F$ is an $L\times K$ matrix and $G$ is an $I\times J$ matrix, it only makes sense to talk about the matrix product $GF$ if $L=J$.

So the matrix dimension rule you learned is really there exactly because only certain functions can be composed. The expression $g\circ f$ only has meaning if the outputs of $f$ are valid inputs for $g$.

 

[Mark44]

So as I'm preparing for finals, I'm wondering:

The multiplication of two matrices is only defined under special circumstances regarding the dimensions of the matrices.

Doesn't that require that compositions of linear transformations are only defined in the same circumstances? I can't imagine not being able to not define a composition of linear transformations, can someone demonstrate this?

If A is an m X n matrix, and B is an n X p matrix, then the product AB is defined, and will be an m X p matrix.

A linear transformation T_A: Rⁿ → R^m takes vectors from Rⁿ and maps them to vectors in R^m. A matrix for T_A will by m X n. Think about how T_B would have to be defined (in terms of its domain and codomain) so that the composition T_A ° T_B would make sense. It might be helpful to use constants for the dimensions.

 

[1MileCrash]

Crystal clear, thanks you two.

 

[blue_raver22]

Yes, the multiplication of two matrices is only defined under certain conditions, specifically when the number of columns in the first matrix is equal to the number of rows in the second matrix. This is known as the matrix multiplication rule and it is essential for performing operations on matrices.

In terms of linear transformations, the same concept applies. The composition of two linear transformations can only be defined when the dimensions of the transformations match up in a certain way. This is because the composition of linear transformations is essentially the multiplication of their corresponding matrices.

For example, if we have a transformation T1 that maps from a 3-dimensional space to a 2-dimensional space and a transformation T2 that maps from a 2-dimensional space to a 4-dimensional space, their composition T1 ∘ T2 would not be defined since the dimensions do not match up (3-dimensional output from T1 and 4-dimensional input for T2).

In terms of demonstration, you can try to compose two linear transformations with mismatched dimensions and see that the resulting transformation is not defined. This highlights the importance of understanding the matrix multiplication rule and its application in linear transformations.