Basic doubts in matrix

Question

Today I was studying about matrices and got stuck with a basic question in my mind . So basically in my book it is written that matrix is a 2 dimensional arrangement of real numbers in rows and columns . But as I had gone through it i realized that matrices are just doing the mathematical problems like that solving linear equations in their own way by arranging the numbers needed to be operated in the problem in a 2D array. They have their own rules to operate on the entities and the output of their operation is also a matrix. Again when we are supposed to rotate the coordinate axes through some angle and them if we want to find out the new position of a point which was let say (2,3) in the previous one. Matrix can solve this by arranging the coordinates in a array and then operating on that matrix by a special matrix which has the property to convert the coordinates into a new one when it's multiplied with it. So can we say that matrices are a new mathematical structures whose rules of operations have been defined by mathematicians? And specifically are matrices an alternative way to solve the problems we used to do normally in a different way ? Like different programming languages in computer

Answer 1

It sounds like you're learning some basic linear algebra, so I'll be writing with that in mind.

A key point that many students (seem to) miss is the difference between a linear transformation and a matrix. Fundamentally, matrices are a useful "bookkeeping" tool for linear transformations. They present all the "information" in the linear transformation in a neat, efficient way.

When you talk about "rotating coordinate axes by some angle", you're talking about a linear transformation of $\mathbb{R}^n$. However, this linear transformation (and every linear transformation, in fact) can be represented by a matrix. That is, for a linear transformation $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$, we can write $T(v) = Av$, where $A$ is an $m \times n$ matrix.

As for matrix operations, mathematicians defined them to be consistent with the underlying linear transformations the matrices represent. Many students feel that matrix multiplication doesn't make sense (why do we go across the rows and down the columns? why not the other way? why not multiply them point by point?). Well, the reason for the seemingly arbitrary definition follows from fact that linear maps can be composed. Given linear transformations $S: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $T: \mathbb{R}^m \rightarrow \mathbb{R}^k$, we can look at their composition $T \circ S: \mathbb{R}^n \rightarrow \mathbb{R}^k$. This transformation is also linear, and defined by $(T \circ S)(v) = T(S(v))$.

In matrix notation, if we write these functions as $S(v) = Av$ and $T(v) = Bv$, then we have $T(S(v)) = T(Av) = BAv$. If we want this to make sense, the product of the matrices $B$ and $A$ had better be consistent with the composition of the functions they represent. And it turns out, the "across the columns, down the rows" matrix multiplication makes this consistent.

Using matrices isn't an "alternative way" to do mathematics. It's just a convenient notation for linear transformations.

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It's worth noting that there are some places where using matrices as a bookkeeping tool is counterproductive. For example, suppose I asked you to prove that matrix multiplication is associative: that is, for matrices $A,B,C$, we have $(AB)C = A(BC)$. If you try to write out the matrices entry by entry, this is a mess! However, if we recall that these matrices represent linear maps, and that function composition is associative, the proof is two lines long.

Answer 2

I'd like to add one thing to @Joe's excellent answer, to try to address the idea that matrices are "new" in some way. Your book's definition of a matrix as a 2D arrangement of numbers is nice and simple, but it requires notions like "2D" and "arrangement". There's a simpler story.

Let me start with vectors: maybe you've been told a vector is a 1D list of numbers, or something like that. But if you've been learning mathematics with various axioms ("addition is commutative", for example), you may think "Hey, we never defined a list!" Let me help out. I'm going to assume you know what a set and a function are, and that you know about the natural numbers ($1, 2, 3, \ldots$ for the sake of this explanation, although many folks start at $0$).

For any natural number $n$, let's let $1:n$ denote the set $\{1, 2, \ldots, n\}$. Then an $n$-vector (or "vector with $n$ entries", or "vector with $n$ components" or "$n$-dimensional real vector") can be defined to be ...a function from $1:n$ to $\Bbb R$.

For instance, the vector that you might write as $b = (1, -2, 4)$ is (in my terms) the function $$ b: \{1,2,3\} \to \Bbb R : i \mapsto \begin{cases} 1 & i = 1 \\ -2 & i = 2\\ 4 & i = 3 \end{cases} $$

You'll notice that (in my world) $b(1) = 1, b(2) = -2, b(3) = 4$, while in your world, you have $b_1 = 1, b_2 = -2, b_3 = 4$. In short, I've replaced "1D arrangement of numbers" with "a particular kind of function" and "subscripting with $i$" by "applying the function to $i$".

If you look carefully at the rules for addition of functions, etc., you'll find that my "function things" behave just like your vectors. And once you do that, you realize that vectors with their new notation (subscripts, parentheses and commas, etc.) are just a repackaging of something you already knew about (i.e., functions).

In the same way, an $n \times k$ matrix is just a function $$ m : (1:n) \times (1:k) \to \Bbb R $$

Some programming languages (Fortran, Matlab) actually use the same notation for subscripts and for function application. I halfway like this (because of the "vectors and matrices are functions" idea presented above), and halfway dislike it, because the implementations of the two things in most languages are wildly different, and the textual similarity may lead one to think that they are somehow the same in terms of things like computational complexity, etc.

Once I've said "Vectors and matrices are just functions," does that mean you should always write $b(i)$ instead of $b_i$? And that you should always write function-like descriptions of vectors rather than just listing their elements like this: (1, -2, 4)? Not at all! Once you know about the correspondence, you should use whatever notation lets you express things most compactly and clearly (for you). For most of us, this involves writing rows or columns of numbers, and rectangular arrays of numbers.