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# A-Level Definition of a Matrix watch

1. I never did A-levels - how does A-Level define a matrix.
I have always defined it as a linear transformation with a basis - I imagine a-level is similar could someone clarify - thanks.
2. (Original post by tombayes)
I never did A-levels - how does A-Level define a matrix.
I have always defined it as a linear transformation with a basis - I imagine a-level is similar could someone clarify - thanks.
A rectangular array of numbers.
3. (Original post by TeeEm)
A rectangular array of numbers.
Yes. I have heard that a matrix is sometimes defined like this. But then how do you explain matrix multiplication? Since clearly, if you define it as a linear transformation with a basis then matrix multiplication is just composition of functions and using that you can get a formula for matrix multiplication trivially. But using your definition how do you explain matrix multiplication?
4. (Original post by tombayes)
Yes. I have heard that a matrix is sometimes defined like this. But then how do you explain matrix multiplication? Since clearly, if you define it as a linear transformation with a basis then matrix multiplication is just composition of functions and using that you can get a formula for matrix multiplication trivially. But using your definition how do you explain matrix multiplication?
I am no purist but define is by definition define
So I define matrix multiplication (an operation) as dotting rows and columns.
How do I define tanx?
I define it as sinx/cosx.
To me both definitions work the same and hence I can sleep at night ...
5. (Original post by TeeEm)
I am no purist but define is by definition define
So I define matrix multiplication (an operation) as dotting rows and columns.
How do I define tanx?
I define it as sinx/cosx.
To me both definitions work the same and hence I can sleep at night ...
no but why does it make sense to do this - you have you justify what your doing - why is it a natural operation?
6. In principle, a matrix is just a table of numbers. That's the definition.

You can define any operation on matrices that you like. And so, we define *multiplication* of matrices in the way that you understand.

If you want to define a slightly different operation - for example one that just multiplies cell-by-cell, you can certainly define that, but that shouldn't now be called "multiplication" and it won't be relevant at A-level. (Aside: this example is sometimes called the "element-wise product" or "Hadamard product" instead, to avoid confusion)

It's useful to define multiplication of matrices in the standard way, for example, because if you are using matrices to represent transformations, then multiplication will have the useful effect of completing the transformations one-by-one. This is what is introduced by A-level.
7. (Original post by tombayes)
Yes. I have heard that a matrix is sometimes defined like this. But then how do you explain matrix multiplication? Since clearly, if you define it as a linear transformation with a basis then matrix multiplication is just composition of functions and using that you can get a formula for matrix multiplication trivially. But using your definition how do you explain matrix multiplication?
We did matrices for O level when I was at school, so we were simply introduced to them as rectangular arrays of numbers and presented with a set of standard rules for adding, subtracting and multiplying matrices of various dimensions with the restrictions that rows and columns needed to be of consistent size when multiplying because you do it "row by column". That was when we were about 12/13 years old!

A year or so later we were taught about the practical uses of matrices e.g. use of 2x2 square matrices to represent simple transformations such as rotation and shear, plus how to write simultaneous equations in 2 variables in matrix/vector form and how to solve them by calculating the inverse of a 2x2 matrix.

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Updated: January 13, 2015
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