Why does Quantum Mechanics use Linear Algebra? I am currently doing Linear Algebra in hopes of one day tackling QM, and I need some motivation now to continue in this pursuit. The University I attend set this as a pre-requisite for QM. Now I have gone as far as Projections Onto Subspaces in LA but I just can't wrap my head around why I might be asked to do the things I am currently doing in my Linear Algebra course for my future work  in QM; my questions are as follows:
-> I have recently checked Shankar's book on QM, and the initial chapters seem to cover Linear Algebra. Over there I have seen him mentioning Linear Operators/Hilbert Spaces/Inner Products etc. I want to know, what do you do in QM that warrants the use of these? What are their physical interpretations?
-> Searching on Google, i have seen that Bra-Ket notations have really to do with vector spaces and dual spaces while operators are links between them, I have seen mentions of Hilbert spaces as well ; I then want to know, how is linear algebra, a seemingly abstract field, interwoven with Physics [the study of natural phenomena]
-> Universities also include a book like Linear Algebra Done Right as a part of their QM curriculum sometimes mainly to do with operators. Now I know operators from Transformations and all, but what has it to do with QM? Why do you need Operators of the form of transformations here? and why linear operators specifically? what's the physical significance of it?
-> Lastly, what role do Eigenvalue problems from my class on LA play later on in QM? Like how might I be expected to interpret and solve them?
PhysSE didn't have a question set like this before, I just wanted to write and get answers collectively and coherently from others way more experienced in this field. While also on Google, I didn't get an exact answer as it was all about online lectures/notes/YouTube recommendations and what not.
I did not get concrete answers, so do not mind please, I am just a beginner looking for motivation to continue with the pre-req's before starting it for real. It is hoped that other freshmen starting LA might also find this thread useful [If you believe some other questions like those above might be useful to keep in the back of my mind at this stage, please comment and share answers as well].
 A: This is quite a broad list of questions, the formal construction of quantum mechanics quite heavily leans on linear algebra. You will have a much easier time learning this topic if you already have a reasonable conceptual understanding of the topics you've listed. Justifying why linear algebra, in particular, happens to provide a good mathematical framework for quantum mechanics is a bit of a philosophical question so I won't address it, but I will outline for you broadly the relationship between linear algebra and quantum mechanics. This will of course by no means be complete or rigorous.
The State Space

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*Given a quantum system, every possible state the system can be in is represented by a vector in an (often infinite-dimensional) Hilbert space, denoted $\mathcal{H}$.

Observable Quantities

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*We associate to every physically measurable quantity (e.g. position, momentum, energy) a self-adjoint operator on this Hilbert space.


*Self-adjoint operators have the special property that their eigenvalues are real and that eigenvectors corresponding to different eigenvalues are orthogonal.
Time Evolution

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*Given a quantum system in a state $|\psi\rangle\in \mathcal{H}$, where we are using bra-ket notation $\bigl(|\psi\rangle$ is just an element of the Hilbert space $\mathcal H\bigr)$, we can ask how the system will evolve in time, this is given by the Schrodinger equation: $$\mathrm i\hbar \frac{\partial}{\partial t}|\psi\rangle=\hat H|\psi\rangle$$ in which $\hat H$ is a self-adjoint operator called the Hamiltonian that, following the prescription in the previous section, corresponds to the physical observable of the energy. That is, the Hamiltonian is a special observable that dictates the time-evolution of a quantum system.

Measurement

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*This is now where we stop talking about pure linear algebra and talk about something physical, everything we've defined above is enough to talk about the process of measurement.


*First, we have to choose a quantity to measure, let's say we measure the position of our system. It is an axiom of quantum mechanics that:



*

*Immediately after measurement, regardless of what state the system was in beforehand, the system will be found in a state that is one of the eigenvectors of whichever self-adjoint operator corresponds to the observable we are measuring.


*Exactly which eigenvector the system "collapses" to after measurement is fundamentally probabilistic, the probability corresponds to the absolute square of the projection of the state before measurement onto the eigenvector (this process is a little more complicated if two eigenvectors correspond to the same eigenvalue).


*The numerical outcome of that measurement will be the eigenvalue corresponding to the eigenvector given in the first point. This is why we restricted our discussion to self-adjoint operators, since their eigenvalues are guaranteed to be real, and we would expect the outcome of measurement to be a real value.



*

*From here the system evolves in time as before, according to the Schrodinger equation.

Wave Functions

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*If you are an undergraduate you will likely be introduced to wave functions rather than Hilbert spaces. I personally wish it were the other way round since the latter is more formal, but that is how it is. The correspondence between the two is that the Hilbert space defined above $\mathcal H$ is isomorphic to the set of square integrable functions on an interval.

There is of course a lot more to quantum mechanics than I have been able to say above, this is intended as a very rough and broad piece of guidance.
A: If you want to study quantum mechanics, keep on working on linear algebra and try to really understand it. To put it short, you describe a quantum mechanical system using a state $|\psi\rangle$, which you pick out of a Hilbert space $\mathcal{H}$ consisting of all possible system configurations. In other words, a state corresponds to a vector out of a complex vector space. From your LA class you probably know it is possible to express the same vector in different bases: in QM you do the same by expressing $|\psi\rangle$ as a linear combination of basis elements out of an orthonormal (as usually states are normalized such that $\langle\psi|\psi\rangle=1$) spanning $\mathcal{H}$. Hermitian operators acting on $\mathcal{H}$ come into play to describe physical observables: a consequence of hermiticity is the eigenvalues being real, indeed the eigenvalues of an operator are the only allowed measurement outcomes for a measurement of the associated physical observable. As a consequence, the corresponding eigenvectors are the only one allowed post-measurement states. I could go on with many others examples, as the fact that the probabilities for measurement outcomes are formulated in terms of the absolute value of scalar products over $\mathcal{H}$, but I would just suggest you to continue studying linear algebra, maybe not only to study quantum mechanics at some later point :)
A: I will try to be as short as possible.

I have recently checked Shankar's book on QM, and the initial chapters
seem to cover Linear Algebra. Over there I have seen him mentioning
Linear Operators/Hilbert Spaces/Inner Products etc. I want to know,
what do you do in QM that warrants the use of these? What are their
physical interpretations?

In quantum mechanics, as the name suggests, things are often quantized. In general, the processes involving quantisation are tackled the best with the use of eigenvalue problems. Schroedinger's time independent equation is an eigenvalue equation, which looks like this:
$$\Big(-\dfrac{\hbar^2}{2m}\nabla^2+V(\vec{r}) \Big)\psi(\vec{r})=E\ \psi(\vec{r})$$
The operator that is being applied on the left hand side is called the Hamiltonian and it is a linear operator. The quantity $\psi (\vec{r})$ is called the eigenfunction of the Hamiltonian, and $E$ is called the eigenvalue corresponding to that eigenfunction.

Searching on Google, i have seen that Bra-Ket notations have really to do with vector spaces and dual spaces while operators are links between them, I have seen mentions of Hilbert spaces as well ; I then want to know, how is linear algebra, a seemingly abstract field, interwoven with Physics[the study of natural phenomena]

You see above, the eigenfunction of the Hamiltonian that I talked about is a vector in Hilbert space, and you need an infinite number of basis vectors to fully span that vector space ( I am assuming you understand this), because no finite numbers are enough. Moreover, it should be clear to you by now how linear algebra is the language of QM, since eigenvalue equation is itself asserting quantised energy values.

Universities also include a book like Linear Algebra Done Right as a part of their QM curriculum sometimes mainly to do with operators. Now I know operators from Transformations and all, but what has it to do with QM? Why do you need Operators of the form of transformations here? and why linear operators specifically? what's the physical significance of it?

I think I have shown you a straightforward example above for this.

Lastly, what role do Eigenvalue problems from my class on LA play
later on in QM? Like how might I be expected to interpret and solve
them?

Since most quantities will be quantised, like the angular momentum, energy etc. you need to know what those quantised values actually are? This is done by solving the eigenvalue problem.
Moreover, there is another way to deal with QM called Matrix Mechanics formulated by Heisenberg, which works entirely on matrices. The Hamiltonian is a matrix, the eigenvector is a matrix and the eigenvalue is a number. So you do really need to know a great deal of LA before you can get a good grip over QM.
A: Hamiltonian is an operator in QM and the most important one. It can be represented as a matrix.
eigenvectors of this hamiltonian matrix will be eigenstates of your system, and eigenvalues of this matrix will be energy levels of your system.
So the eigenvalue problem is the most fundamental and important starting point in QM, the whole QM is built around this. And if you take a QM course this will be mostly the only thing to do. you will work with different hamiltonians for different systems, and you will solve eigenvalue problems and you will see the physical implications. Thus, undergrad QM is just applied linear algebra.
A: I suggest you read about some of the history of QM. Heisenberg originally came up with square arrays of numbers for the position $X$ and momentum $P$ of particle in a harmonic oscillator. He discovered  that there was some funnny way of getting a similar  square array for their product $XP$. A mathematician friend (Max Born) told him that these  arrays were called "matrices" and his "funny way"  was called matrix multiplication and had been studied quite a bit in the 19th century. It all started from there.
