Matrices Intuition

Question

I am currently studying matrix algebra. The axioms and theorems of this form of algebra are a bit different from the high school algebra I did. However one knows that one is dealing with real numbers in that form of algebra and one can always associate a physical object with a number to see if a particular theorems makes sense. For example, x-2=0 , find x. Now one can associate 2 and x with apples and see the operation '-' as eating apples and then having 2 apples and eaten x apples you get 0 apples, hence x must be 2 (apples) in this case.

Similarly is there anything physical that I can associate with a matrix so that I can call upon my intuition whenever confronted with a new theorem/result instead of blindly following algebra rules and axioms associated with a matrix.

Any help in this regard will be appreciated.

In short: Please associate a common physical object that obeys all the rules of algebra of a matrix.

Answer 1

Not necessarily a physical object, but but linear algebra is all about vectors. Almost all operations and theorems have something to do with vectors. A vector could be thought of as velocity, momentum, force, moment, a field, etc.

Answer 2

Let's say you wanted to solve eleven equations: $x+y=1$, $x+y=2$,..., $x+y=11$. Using matrices you could write this as

$\left[\begin{array}{cc} 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1\\ 1 & 1 \end{array}\right]\left[\begin{array}{c} x\\ y \end{array}\right]=\left[\begin{array}{c} 1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\\ 9\\ 10\\ 11 \end{array}\right] $

If you write out the result of this matrix equation you will get the 11 equations from that you wanted to solve.

So as a starting intuition you can think of matrices as spreadsheets that help you keep track of a lot of equations. The algebra rules are just how you can manipulate your spreadsheet without bonking it up. The fact that $\bf{AB\neq BA}$ well spreadsheets don't work that way, try reversing the order of the example above.

If you want to take it further, you can look at a matrix as a machine that takes objects like apples, eggs and shoes, and combines them. But you also want try several different combinations of apples, eggs and shoes, that's where the spreadsheet like nature of matrices help out.

So let's say you want combine 1 apple with 2 eggs and 3 shoes and then you want to record what you get. After that, try combining 3 apples with 2 eggs and 1 shoe and record the result. Also check what just one of each combined gives.

If you let $x,y$ and $z$ represent apples, eggs and shoes respectively, in matrix algebra you get

$\left[\begin{array}{ccc} 1 & 2 & 3\\ 3 & 2 & 1\\ 1 & 1 & 1 \end{array}\right]\left[\begin{array}{c} x\\ y\\ z \end{array}\right]$

So you can think of the matrix as a machine that takes a vector of objects as inputs, and then outputs a new vector with the resulting combination. In our case our output is

$\left[\begin{array}{c} x+2y+3z\\ 3x+2y+z\\ x+y+z \end{array}\right]$

Notice that this output is one column just like $\left[\begin{array}{c} x\\ y\\ z \end{array}\right]$.

So I want to find out what I get when I combine 1 apple, 2 eggs and 3 shoes, but when I mess around with the abstract equation $\bf{A}\vec{x}=\vec{y}$, I don't want to end up combining say 2 apples, 4 eggs and 1 shoe instead. The rules and axioms ensure that no matter what I do with an abstract equation, I don't change what I'm actually trying to accomplish with the matrix.