# What is the difference between matrix theory and linear algebra?

I have lifted this from Mathoverflow since it belongs here.

Hi,

Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What is the difference, if any, between matrix theory and linear algebra?

Thanks! kolistivra

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Would you be okay with me quoting my answer from MO? –  Qiaochu Yuan Jul 28 '10 at 18:06
Of course. You can quote others answers too. I have no objections whatsoever. . :D –  user218 Jul 28 '10 at 18:15
This question received satisfying answers on mathoverflow and I see no reason for it to be here. –  Eric O. Korman Jul 28 '10 at 19:01

A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another. You can do lots of interesting numerical things with matrices, and these interesting numerical things are very important because matrices show up a lot in engineering and the sciences.

In linear algebra, however, you instead talk about linear transformations, which are not (I cannot emphasize this enough) a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. The difference between a linear transformation and a matrix is not easy to grasp the first time you see it, and most people would be fine with conflating the two points of view. However, when you're given a linear transformation, you're not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you're only allowed to ask for things that don't depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.

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I think of Linear Algebra is a semi-applied Matrix theory. I can't summarize it better then Qiaochu. –  Tyler Hilton Jul 28 '10 at 18:21
Nice answer! Questions: (1) Are matrices always associated with vector spaces? Can they connect to the outside of vector spaces? (2) Is linear algebra the theory of vector spaces, or theory of both vector spaces and of matrices? –  Tim Aug 21 '11 at 23:14
@Tim: 1) Yes. I would argue that if you are using a matrix in a context where there isn't some implicit vector space attached, you shouldn't be calling it a matrix, but a 2-dimensional array. 2) Linear algebra is the theory of vector spaces and linear transformations between them (which are matrices in a particular choice of basis). –  Qiaochu Yuan Aug 21 '11 at 23:36