Is the solution to a 2D system of equations a point or a vector?

Question

I'm reading a Linear Algebra text and an introductory example they give is to solve the system of equations:

x + 2y = 3
x - y = 0

When thinking of this as two (possibly) intersecting lines, the solution is easy to solve for and think of geometrically as the point (1, 1).

However, when thinking of this in terms of the matrix representation Ax = b, where A = $\begin{bmatrix} 1 & 2 \\ 1 & -1 \\ \end{bmatrix}$, x = $\begin{bmatrix} x \\ y \\ \end{bmatrix}$, and b = $\begin{bmatrix} 3 \\ 0 \\ \end{bmatrix}$, writing the solution as the vector x = $\begin{bmatrix} 1 \\ 1 \\ \end{bmatrix}$ confuses me a bit. The reason being because I envision vectors as having magnitude and direction, as well as being things we can pick up and draw anywhere in our space (like how the wind could be blowing 5mph to the East in two different places and we'd use the vector $\begin{bmatrix} 5 \\ 0 \\ \end{bmatrix}$ to represent the wind in both places).

I understand that using matrix notation and vector notation allows us to row-reduce the augmented matrix and get the solution, but isn't the solution just the point (1, 1)? Or is there some special geometric reason we say the answer is the solution vector x = $\begin{bmatrix} 1 \\ 1 \\ \end{bmatrix}$? Is it through the lens of not considering the system as two intersecting lines but as a transformation of sorts that puts in the vector x = $\begin{bmatrix} 1 \\ 1 \\ \end{bmatrix}$ and pops out the vector b = $\begin{bmatrix} 3 \\ 0 \\ \end{bmatrix}$?

Thanks for the clarifications in advance!

Answer 1

For me, I interpret the x and b not as physical/geometric vectors, but as mathematical/numerical vectors, which are just collections of numbers. Like an array in programming language. For simplicity, people use the same [] notation for both physical and mathematical vectors because the general consensus is that it's not worth the effort to distinguish, and that if you really want to know, it's expected to be implied from the problem context.

It seems the distinguishing features of a physical vector are special to you, so I recommend viewing these vectors in linear algebra systems as numerical arrays by default.

That being said, one can choose to interpret the same vectors as physical vectors if you establish more context for the problem; e.g. that x and b are vectors with origin $(0,0)$ and $A$ is a transformation matrix that scales and rotates vector x into vector b. While physically the original and context-added problem are different, mathematically they're equivalent (one of the beauties of maths is that it allows you to compare two seemingly different contexts). If you're interested in how to interpret $A$ as a transformation matrix, I recommend this video by 3Blue1Brown.

Answer 2

This depends entirely on your interpretation of the problem the equations are part of. Personally, I wouldn't have used matrices to solve these equations- that seemed to me to be "over-kill". Subtracting these equations, x is eliminated and 3y= 3 so y= 1. Then x- y= x- 1= 0 so x= 1.

Now, I can interpret that as simply two numbers, x=1 and y= 1. Or I can interpret it as a point, (1, 1), in a Cartesian coordinate system. Or I can interpret as a vector, i+j. How I would interpret it depends on how the problem was worded. If there were only the equations with nothing else, I would probably interpret the answer as two numbers, but that's just me!