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\begin{matrix} A \in R^{3 \times 3} \\ P L U decomposition of A : \\ P = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}), L = (\begin{matrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 1 & 1 & 1 \end{matrix}), U = (\begin{matrix} 2 & - 3 & 2 \\ 0 & 3 & 1 \\ 0 & 0 & 7 \end{matrix}) \\ Solve A x = (\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}) without calculating A \end{matrix}

\begin{matrix} A \in F^{n \times n}, λ \in F \\ λ is called an eigenvalue if \\ \exists v \neq 0 \in F^{n} : A v = λ v \\ 1. Prove: V_{λ} = {v \in F^{n} | A v = λ v} is a vector subspace of F^{n} \\ 2. Prove: \exists v, w \in F^{n} : A v = λ v, A w = μ w, λ \neq μ ⟹ {v, w} is a linear independence \\ 3. Prove: If A is nilpotent, then 0 is it’s eigenvalue \\ 4. Prove: If A is nilpotent, then 0 is it’s only eigenvalue \\ 5. Prove: 0 is an eigenvalue of A ⟺ Set of columns of A is a linear dependence \end{matrix}