Register to reply

Very basic linear algebra question

by 1MileCrash
Tags: algebra, basic, linear
Share this thread:
1MileCrash
#1
Dec5-13, 01:57 PM
1MileCrash's Avatar
P: 1,302
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?
Phys.Org News Partner Mathematics news on Phys.org
Heat distributions help researchers to understand curved space
Professor quantifies how 'one thing leads to another'
Team announces construction of a formal computer-verified proof of the Kepler conjecture
economicsnerd
#2
Dec5-13, 02:16 PM
P: 239
If [itex]f:V\to W[/itex] and [itex]g:X\to Y[/itex] are linear transformations, it only makes sense to talk about the composition [itex]g\circ f[/itex] if [itex]W=X[/itex]. In particular, if [itex]f:\mathbb R^K \to \mathbb R^L[/itex] and [itex]g:\mathbb R^J \to \mathbb R^I[/itex] are linear transformations, it only makes sense to talk about the composition [itex]g\circ f[/itex] if [itex]\mathbb R^L=\mathbb R^J[/itex], i.e. if [itex]L=J.[/itex]

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

So the matrix dimension rule you learned is really there exactly because only certain functions can be composed. The expression [itex]g\circ f[/itex] only has meaning if the outputs of [itex]f[/itex] are valid inputs for [itex]g[/itex].
Mark44
#3
Dec5-13, 02:17 PM
Mentor
P: 21,321
Quote Quote by 1MileCrash View Post
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 TA: Rn → Rm takes vectors from Rn and maps them to vectors in Rm. A matrix for TA will by m X n. Think about how TB would have to be defined (in terms of its domain and codomain) so that the composition TA TB would make sense. It might be helpful to use constants for the dimensions.

1MileCrash
#4
Dec5-13, 02:23 PM
1MileCrash's Avatar
P: 1,302
Very basic linear algebra question

Crystal clear, thanks you two.


Register to reply

Related Discussions
Linear Algebra: basic proof about a linear transformation Calculus & Beyond Homework 3
Basic question of linear algebra Linear & Abstract Algebra 6
Basic Linear Algebra Proof Calculus & Beyond Homework 3
Help with basic linear algebra Calculus & Beyond Homework 2
Basic linear algebra Calculus & Beyond Homework 16