I know this question has been asked many times over here, but I never found something that really matches my requests and my needs, so I thought of asking a question in a different way: I will write down here three examples of exercises our professor gave on past exams, in order for you to fully see what I am dealing with. $\DeclareMathOperator{\tr}{tr}$
From there, my question is the same: can you recommend me a great book with which I could learn these topics in a very rigorous and good way? Thank you!
Ex. 1
Be $V$ a vector space over $\mathbb{Q}$ and be $f \in \text{end}(V)$ an endomorphism with characteristic polynomial $p_f(x) = x^3-3x^2+2x$.\ Calculate the characteristic polynomial of $g = f + (f + 2I)^{-1}$ and determine algebraic and geometric multiplicity of every eigenvalue of $g$. How does the result change if $V$ is a vector space over $\mathbb{Z}/5\mathbb{Z}$?
Ex.2
Over the space $V = M_n(\mathbb{R})$ consider the symmetric bilinear form $\beta(A, B) = \tr(AB) -\tr(A)\tr(B)$. Be $\mathcal{A}$ and $\mathcal{B}$ respectively the subspace of $V$ of the antisymmetric and traceless symmetric matrices, and be $L= \text{span}\{ I_n \}$.
Show $\mathcal{A}, \mathcal{B}, L$ are orthogonal wrt to $\beta$, find the signature of $\beta$.
Ex. 3
Be $V = \mathbb{Q}^3$ and $W_1 = \left \{ \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \in \mathbb{K}^3:\ x_1 -2x_2 + x_3 = 0,\ 2x_1 - x_2 + 2x_3 = 0 \right\}$ and $W_2 = \text{span} \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \right \}$.
Say if it does exist a basis $\{ b_1, b_2, b_3 \}$ of $V^*$ such that
$$b_1 \in \text{ann}(W_1 + W_2) \qquad b_2 \in \text{ann}(W_1) \qquad b_3 \in \text{ann}(W_2)$$ and in case build it.
Thank you!
As I said in my comment, this post seems helpful. The pdf linked in that question seems a bit focused on $\Bbb R^n$, in contrast to the field-agnostic nature of your first and third questions. As far as popular textbooks are concerned, Artin's Algebra seems like a good candidate.
While I'm at it, I'll give my approach for the second question. Every symmetric bilinear form over $M_n(\Bbb R)$ can be uniquely represented in the form $$ \beta(A,B) = \langle A,\phi(B)\rangle, $$ where $\langle A,B \rangle = \operatorname{tr}(A^TB) = \operatorname{tr}(AB^T)$ is the Frobenius inner product and $\phi:M_n(\Bbb R) \to M_n(\Bbb R)$ is a linear map. Note that $\phi$ is necessarily self-adjoint and the signature of $\beta$ can be ascertained from the signs of the eigenvalues of $\phi$.
For the $\beta$ given in the question, the associated map $\phi$ is $\phi:X \mapsto X^\top - \operatorname{tr}(X)I$. Indeed, we can verify that this is the case by seeing that $$ \langle A,\phi(B) \rangle = \langle A, B^T - \operatorname{tr}(B)I \rangle = \langle A,B^T \rangle - \operatorname{tr}(B)\langle A,I \rangle = \operatorname{tr}(AB) - \operatorname{tr}(A)\operatorname{tr}(B). $$ Relative to this linear map, note that $\mathcal L$ is the eigenspace of $\phi$ associated with the eigenvalue $1 - n < 0$, $\mathcal A$ is an eigenspace with associated eigenvalue $-1$, and $\mathcal B$ is an eigenspace with associated eigenvalue $1$. With that, the total multiplicity of the positive eigenvalues is $$ n_+ = \dim(\mathcal A) = n(n-1)/2, $$ and the total multiplicity of the negative eigenvalues is $$ n_- = \dim(\mathcal L) + \dim(\mathcal A) = 1 + n(n-1)/2 = n(n+1)/2. $$