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BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Clarify the theory of bases and dimensions for the linear subspaces [math]V\subset\mathbb R^N[/math], notably by establishing the formula

[[math]] \dim(\ker f)+\dim(Im f)=N [[/math]]

valid for any linear map [math]f:\mathbb C^N\to\mathbb C^N[/math], and then extend this into a theory of abstract linear spaces [math]V[/math], which are not necessarily subspaces of [math]\mathbb C^N[/math].

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Work out what happens to the main diagonalization theorem for the matrices [math]A\in M_N(\mathbb C)[/math], in the cases [math]A\in M_2(\mathbb C)[/math], [math]A\in M_N(\mathbb R)[/math], and [math]A\in M_2(\mathbb R)[/math].

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Establish with full details the resultant formula

[[math]] R(P,Q)= \begin{vmatrix} p_k&&&q_l\\ \vdots&\ddots&&\vdots&\ddots\\ p_0&&p_k&q_0&&q_l\\ &\ddots&\vdots&&\ddots&\vdots\\ &&p_0&&&q_0 \end{vmatrix} [[/math]]

and discuss as well what happens in the context of the discriminant.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Clarify which functions can be applied to which matrices, as to have results stating that the eigenvalues of [math]f(A)[/math] are [math]f(\lambda_1),\ldots,f(\lambda_N)[/math].

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Work out specialized spectral theorems for the orthogonal matrices [math]U\in O_N[/math], going beyond what has been said in the above.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Prove that any matrix can be put in Jordan form,

[[math]] A\sim\begin{pmatrix} J_1\\ &\ddots\\ &&J_k \end{pmatrix} [[/math]]

with each of the blocks which appear, called Jordan blocks, being as follows,

[[math]] J_i=\begin{pmatrix} \lambda_i&1\\ &\lambda_i&1\\ &&\ddots&\ddots\\ &&&\lambda_i&1\\ &&&&\lambda_i \end{pmatrix} [[/math]]

with the size being the multiplicity of [math]\lambda_i[/math].