Trying to conceptualize vectors in $4$-dimensions. Is this example accurate?

Question

I'm new to learning Linear Algebra, and I'm trying to visualize how a vector in $4$ dimensions may work.

For this, I'm imagining a Soduku puzzle. It is a grid with $3\times 3$ big squares, with each square containing a smaller grid of $3\times 3$ squares.

Could we represent individual boxes as a 4D vector, where we use two dimensions to refer to the big box, and then another two dimensions to refer to the smaller box within the big box?

For example, the top left small box would be $[0,0,0,0]$. The bottom right would be $[2,2,2,2]$.

Am I on the right track with understanding this at all?

If this is ok, could the magnitude, as a scalar, then be used to describe the contents of the box?

For example:

$$v= [0,0,0,0]$$

$$||v|| = 3$$

I know I may be way off target, and I am grateful for your patience. I just don't want to misunderstand so fundamental.

Answer 1

For the purposes of linear algebra, you can get away with visualising everything as either $\mathbb R^2$ or as $\mathbb R^3$. Any intuitive geometric argument (e.g. why projection is defined the way it is) in any higher dimension $n$ can be visualised as if $n = 2$ or $3$. When you aren't visualising anything but just computing, you can treat it as a list of four numbers.

Answer 2

I suppose you could represent a grid like that, but I don't think its very useful. In your example, its essentially a 9x9 grid, so you only need two dimensions to specify any one square (vertical and horizontal). If you imagine a column of 9 cubes coming out of every square, you would have a 9x9x9 cube so you would need 3 dimensions to specify any one cube. We cannot imagine the 4th dimension but it is essentially the same idea, we would get a 9x9x9x9 hypercube and would need 4 dimensions to specify any one of the 4 dimensional cubes.

When dealing with higher dimensions, you have to let go and just follow the math.