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I'm currently a beginner at linear algebra. So, in some books I see authors start defining linear equations and then they define matrices and, supposedly, the definition of associative matrix is to handle linear equations easily. However they never establish the connection between both objects and never explain why it is possible to work with matrices in substitution of linear equations.

For some classmates this is irrelevant because they say that I just complicate my life with such questions. But it is important and think that the treatment given in such books is either very informal so beginners like me can understand the concepts or maybe is too simple that I'm missing something. I have read that two objects are generally treated as being the same, of course under certain properties, if there is a connection between them in terms of a one-to-one correspondence (something called morphism, isomorphism, monomorphism, etc).

So, how would you establish the bijection between linear equations and matrices considering elementary operations?

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1 Answer 1

The general form of row $i, \, 1 \leq i \leq m$, of a system of $m$ linear equations in $n$ variables $x_j$ can be written as $$\sum_{j=1}^n a_{i,j} x_j = b_i,$$ $a_{i,j}, b_i \in \mathbb{R}.$

The general form of row $i$ of an $m \times n$ matrix $A$ can be written as $$(a_{i,1}, \dots, a_{i,n});$$ the general form of a vector $b \in \mathbb{R^m}$ as $${(b_1, \cdots, b_m)}^t.$$ You are looking at the matrix equation of form $$Ax = b,$$ for the same values $a_{i,j}, b_i$ as in the system of equations.

Let $E$ be the space of such equations for $b_i == 0$ (as we only map to the matrix, the LHS of the matrix equation), described by its entries $a_{i,j}$, and $M^{m,n}$ be defined in the usual way. So there is a natural bijection $\phi$ between the two, mapping
$$\phi: E \rightarrow M^{m,n}, \quad \{a_{1,1}, \dots, a_{1,n}; a_{2,1}, \cdots , a_{m,n} \} \mapsto A = (a_{i,j}).$$ It's probably obvious that this map is a bijection (also an isomorphism) between the two spaces. Is this what you were looking for? Or was it that applying elementary matrices to a system represented by a matrix equation is the same as performing row-operations on the system, and that these operations do not change the solution space? Showing that is not hard, but would take a fairly longish series of lemmata.

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The biyection between both objtects is easy to stablish, even by using the definition of assosiative matrix. And yes my question was more on the sense that using elementary operations on matrices leads to the same result as treating the system itself. So, could you give me a sketch or books where to look for it? And also could you give me your opinion about why authors omit doing this? –  Daniela Diaz Feb 17 '13 at 23:50
    
@DanielaDiaz: Books: Artin, Algebra, bases his entire approach early on on showing how to use elementary matrices to reduces a matrix $A$ to the so-called row-echelon form, and how this amounts to simplifying systems of equations without changing the solution space. The book will be classified as 'advanced undergraduate' as it continues to introduce the standard algebra and linear algebra undergraduate curriculum for a math major, but this is in chapter 1, and I find it very accessible. I have no other good recommendation. –  gnometorule Feb 17 '13 at 23:58
    
@DanielaDiaz: As to why this is not done more by math teachers, I don't know. I find Artin's approach impeccable in this sense, and in my math background too it was less emphasized. My linear algebra class way back didn't do this either (it was all morphisms), but a class in numerical methods you have to take in your first 2 years at my old school covered this in the context of the Gauss-Jordan-Algorithm. I agree that making this more explicit is very useful, but I'm not a math teacher. :) –  gnometorule Feb 18 '13 at 0:01
    
Thank you so much. I just found the book in my library and, well, it doesnt explain the morphism between the object per se, but it made me think that it is probably unecessary if I start defining matrices and the operation on them first. So trying to solve a system of equations would be just a consequence trying to solve a matrix equation in the form $Ax=B$ being A the associative matrix, X the matrix of unknowns and B the matrix of constants. So I'm going to try this new insight and see what happen. –  Daniela Diaz Feb 18 '13 at 0:35
    
@DanielaDiaz: I think that's the right way to think about this. Good luck! –  gnometorule Feb 18 '13 at 0:38

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