What's the relationship between linear transformations and systems of equations?

Question

I began watching Gilbert Strang's lectures on Linear Algebra and soon realized that I lacked an intuitive understanding of matrices, especially as to why certain operations (e.g. matrix multiplication) are defined the way they are. Someone suggested to me 3Blue1Brown's video series (https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) and it has helped immensely. However, it seems to me that they present matrices in completely different ways: 3Blue1Brown explains that they represent linear transformations, while Strang depicts matrices as systems of linear equations. What's the connection between these two different ideas?

Furthermore, I understand why operations on matrices are defined the way they are when we think of them as linear maps, but this intuition breaks when matrices are thought of in different ways. Since matrices are used to represent all sorts of things (linear transformations, systems of equations, data, etc.), how come operations that are seemingly defined for use with linear maps the same across all these different contexts?

Answer 1

Not sure if I am addressing what you really are after, but I always like considering simple examples if there is something I don't understand. Using the same matrix in two different contexts. Here is an example of a linear transformation. $$ A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} $$ If you input some vector, say $[x_1, x_2] = [1, 1]$, it transforms it by stretching it out in the $x_1$ direction into $[2, 1]$.

The same matrix is used in a related linear equation. $Ax = [2,1]$. What $x$ values gave the stretched vector $[2,1]$? $$ Ax = [2,1] $$ In matrix form. $$ \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} $$ As a system of linear equations. $$ 2x_1 + 0 x_2 = 2 \\ 0x_1 + x_2 = 1 $$

Answer 2

First of all, simply by using matrix multiplication, you can rewrite $$Ax$$

into some equations

$$a_{11}x_1+\cdots+a_{1n}x_n$$

$$\cdots$$

$$a_{m1}x_1+\cdots+a_{mn}x_n$$

This means, that a matrix does nothing more than being a representation of the coefficients of these linear equations, which is more convenient to write down.

If you think of it in a more abstract way, you could even just write down

$$a_{11},\ldots, a_{1n},a_{21},\ldots,a_{2n},\ldots,a_{m1},\ldots,a_{mn}$$

into one vector or text file, where $a_{ij}$ are just some variables for anything. The total amount of variables is $m\cdot n$.

Now, if you have given a matrix $A$ and you vary $x$ you can think of $A$ as a linear transformation of $x$. For example the matrix

$$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$$

will map the plane $\mathbb{R}^2$ onto a plane which has a $45°$ angle with the $x_1$-axis. So in this case it might be better to think of $A$ as a map, which transforms $x$ in a linear way.

However, if you want to think of systems of linear equation with a given $b$, you can rewrite $$Ax=b$$

into some equations

$$a_{11}x_1+\cdots+a_{1n}x_n=b_1$$

$$\cdots$$

$$a_{m1}x_1+\cdots+a_{mn}x_n=b_m$$

Now, you are more likely to have an interest in not transforming the $x$, but finding an $x$ that is a solution to the equations. In this case you are more likely to think of a system of equations, which needs a solution.

Answer 3

You ask

Since matrices are used to represent all sorts of things (linear transformations, systems of equations, data, etc.), how come operations that are seemingly defined for use with linear maps the same across all these different contexts?

Other answers and comments address the connection between linear transformations and systems of equations. You can turn questions about systems of equations into questions about their matrices of coefficients and then about linear transformations, and vice versa. The sum and product operations on matrices correspond to addition and composition of linear transformations. That back and forth illuminates both areas.

Abstractly, a matrix is just a rectangular array of numbers. That can be a useful abstraction in contexts other than systems of equations. It's the underlying abstraction in a spreadsheet. For example, the percentages of urban, suburban and rural areas by state will be a $50 \times 3$ matrix. In that context matrix addition is not likely to be useful.

Answer 4

Consider the system of equations

$$ Ax = b $$

In this system, $A$ is a linear transformation. When that transformation is applied to $x$ it equals $b$. In that sense, it is a system of equations.

For all of the contexts you mention, what is described is always a system of equations for which the value on one side ($b$) is the linear transformation ($A$) of some other quantity ($x$).