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Today I had my first class of a QFT course, and there were some things that apparently I am supposed to know, but I don't. One of them is regarding Lorentz transformations. My teacher stated that:

$\Lambda^{\mu \nu}$ is a Lorentz transformation if: $$x'^\mu=\Lambda^{\mu\nu}x_\nu $$

And honestly I am not sure how to begin understanding this. I know that $\Lambda $is the Lorentz transformation, and that the superscripts indicate the dimensions (like 2x3 or something like that). I remember from SR that the covariant and contravariant forms of four-vectors are related by $x^\nu=-x_\nu$, but I just feel lost as to what that equation means. Could somebody explain it to me or suggest some bibliography from which to catch up?

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    $\begingroup$ Did your QFT course have a prereq that you didn’t take? Did you ever study matrices in a linear algebra course? Do you know how to write a Lorentz transformation without matrices? $\endgroup$ – G. Smith Aug 14 at 0:35
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    $\begingroup$ Are you sure your teacher didn’t write $x’^\mu=\Lambda^\mu{}_\nu x^\nu$? That’s the usual way. $\endgroup$ – G. Smith Aug 14 at 0:39
  • $\begingroup$ Sorry mon but you're deep in it if you can't get past that equation... $\endgroup$ – ZeroTheHero Aug 14 at 1:33
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    $\begingroup$ you really should reread all you SR notes and make sure you understand every single equation on it, also understand the demonstration of why it is valid.What are the assumption, and what are the conclusions $\endgroup$ – Wolphram jonny Aug 14 at 2:06
  • $\begingroup$ This is just matrix multiplication with vector (linear algebra) $\endgroup$ – Eli Aug 14 at 6:44
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$Λ$ is a $ 4\times4$ matrix with the matrix elements $Λ^{\mu\nu}$ so $\mu,\nu=1,2,3,4$. When $Λ$ acts on a four-vector $x$ it produces a four-vector $x'$ $$x'=Λx$$ If we write the same thing but for each component $x'^\mu$ of $x'$ individually then we get $$x'^\mu=Λ^{\mu\nu}x_\nu$$ where, by convention, summation over all matching indices is assumed -- in this case over $\nu$. The meaning of the position of indices -- upper or lower-- has to do with the fact that these are not vectors and matrices in the usual sense but these are tensors.

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