Linear-1 4

LU decomposition #definition

\begin{matrix} A x = b \\ L U x = b \\ U x = \tilde{x} \\ L \tilde{x} = b \end{matrix}

\begin{matrix} P - permutation matrix \\ P^{- 1} = P^{T} \\ P A = L U \end{matrix}

\begin{matrix} V is called vector space if: \\ v \in V, α \in F \\ v_{1} + v_{2} \in V \\ α v_{1} \in V \\ v_{1} + v_{2} = v_{2} + v_{1} \\ α v_{1} = v_{1} α \\ (v_{1} + v_{2}) + v_{3} = v_{1} + (v_{2} + v_{3}) \\ α (β v) = (α β) v \\ \exists 0_{V} : \\ v + 0_{V} = v \\ v + (- v) = 0_{V} \\ \exists 1_{F} : \\ 1_{F} \cdot v = v \\ α (v_{1} + v_{2}) = α v_{1} + α v_{2} \\ (α + β) v = α v + β v \end{matrix}

\begin{matrix} \begin{matrix} u_{1}, u_{2} \in U, α \in F \\ 1. & v_{1} + v_{2} \in U \\ 2. & α v_{1} \in U \end{matrix} ⟹ U is a field \end{matrix}