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I'm learning about tensors in the context of special relativity, and I'm a bit confused some notation.

I understand a four-vector is a four dimensional vector, which is written in the form $(ct, x, y, z)$, in the convention I am using. Sometimes, we refer to the contravariant components of the four vector $x^\alpha$. My understanding is sort of off here, though. Sometimes, we write

$x^\alpha = \Lambda_\beta^\alpha x^\beta$.

I don't really understand what this expression means. On the one hand, I think of this as essentially a matrix multiplication equation, where we have that the $\alpha$'th component $x^\alpha$ of a four vector $\textbf{x} = (x^0, x^1, x^2, x^3)$, is given (writing the explicit sum) as $\sum_{\beta = 0}^3\Lambda_\beta^\alpha x^\beta$.

I have also seen it written that $x^\alpha = (ct, x, y, z)$, which confuses me, since I understood $x^\alpha$ to be a component rather than a vector itself. Though, if we understand $x$'s with superscripts to be vectors, then what could $\Lambda_\beta^\alpha x^\beta$ possibly mean? Given that there is an implied summation over $\beta$, it doesn't make sense to me that $x^\beta$ could be a vector, and not just a component.

On the other hand, I've also heard that greek letter superscripts can be thought of as meaning "in this coordinate system", meaning $x^\alpha$ is a four vector -- not just a component -- in a coordinate system labelled $\alpha$, $x^\beta$ is the coordinates of the same vector in a coordinate system labelled $\beta$, and $\Lambda_\alpha^\beta$ actually $\textit{is}$ a matrix, and not just an entry in a matrix.

I have a similar confusion with the Kronecker delta. I always understood that the Kronecker delta is a function $\mathbb{N^2} \to \{0, 1\}$, defined as

$\delta_{ij} = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}$

It seems like with tensors, it's the same thing, apart from we write $\delta_i^j$, for some reason. I understand that superscripts are for contravariant components and subscripts are for covariant components, but I have no idea why this matters for a function which can only be either 0 or 1. Surely, no matter what $i$ and $j$ are, the end result is the same, regardless of how high up the $\delta$ we've chosen to write the indices?

On the same line of some reading, I have read that

$\delta_i^j = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}$ AND that $\delta_i^j$ equals the identity matrix. How can the same symbol mean two things?

I get that the matrix whose (i,j)th entry is $\delta_i^j$ would be the identity matrix, but surely the function $\delta_i^j$ isn't a matrix itself? I just get really confused when the same symbol means a bunch of different things! Also, if $\delta_i^j$ is thought of as a matrix, assuming that its subscript index tells us the column and the superscript index tells us the row, is this not assigning some sort of different contra/co variance between rows and columns?

So in general, I am really confused about all of this notation I've seen. Can anyone help?

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I think you've pretty much answered the question yourself. Notations like $x^\alpha$ and $\Lambda^\alpha{}_\beta$ can be read either as a notation for a whole tensor or as a notation for one of its components. If you like, you can imagine this as two different notations where the translation between the two notations is trivial.

One thing that seems missing from what you said is the distinction between abstract index notation and concrete index notation. Most relativists today use greek indices for concrete indices, which, as you say, have meaning in a certain coordinate system. But they use latin indices as abstract indices, meaning that the expressions they're writing aren't in any particular coordinate system, and would be valid in any coordinate system.

Also, it's important in general to keep the order of indices straight, so rather than writing things like $a^j_i$, you want to write something like $a^j{}_i$. This distinction is only irrelevant if the tensor is symmetric.

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