Vector spaces

Crazy Vector Space

𝑪𝑽𝑺 = {[\begin{matrix} x \\ y \end{matrix}] ∣ x, y \in ℝ} / ℝ

Operations

\begin{matrix} [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] \oplus [\begin{matrix} x_{2} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{1} + x_{2} - 2 \\ y_{1} + y_{2} \end{matrix}] \\ α ⊙ [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} α x - 2 α + 2 \\ α y \end{matrix}] \end{matrix}

{\vec{e}}_{1} = (\begin{matrix} 1 \\ 0 \end{matrix}) {\vec{e}}_{2} = (\begin{matrix} 0 \\ 1 \end{matrix})

{\vec{b}}_{1} = (\begin{matrix} 5 \\ 9 \end{matrix}) {\vec{b}}_{2} = (\begin{matrix} 4 \\ 7 \end{matrix})

{\vec{e}}_{1} = (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}) {\vec{e}}_{2} = (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}) {\vec{e}}_{3} = (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})

{\vec{b}}_{1} = (\begin{matrix} 0 \\ 2 \\ 1 \end{matrix}) {\vec{b}}_{2} = (\begin{matrix} -1 \\ 3 \\ 2 \end{matrix}) {\vec{b}}_{3} = (\begin{matrix} 1 \\ -2 \\ -1 \end{matrix})

{\vec{d}}_{1} = 1 {\vec{d}}_{2} = x {\vec{d}}_{3} = x^{2}

{\vec{b}}_{1} = x^{2} - 8 x + 5 {\vec{b}}_{2} = x^{2} - 5 x + 3 {\vec{b}}_{3} = - x^{2} + 7 x - 4

(\begin{matrix} 5 & 4 & 1 & 0 \\ 9 & 7 & 0 & 1 \end{matrix}) \overset{}{\to} (\begin{matrix} 1 & 0 & -7 & 4 \\ 0 & 1 & 9 & -5 \end{matrix})

(\begin{matrix} 0 & -1 & 1 & 1 & 0 & 0 \\ 2 & 3 & -2 & 0 & 1 & 0 \\ 1 & 2 & -1 & 0 & 0 & 1 \end{matrix}) \overset{}{\to} (\begin{matrix} 1 & 0 & 0 & 1 & 1 & -1 \\ 0 & 1 & 0 & 0 & -1 & 2 \\ 0 & 0 & 1 & 1 & -1 & 2 \end{matrix})

(\begin{matrix} 5 & 3 & -4 & 1 & 0 & 0 \\ -8 & -5 & 7 & 0 & 1 & 0 \\ 1 & 1 & -1 & 0 & 0 & 1 \end{matrix}) \overset{}{\to} (\begin{matrix} 1 & 0 & 0 & 2 & 1 & -1 \\ 0 & 1 & 0 & 1 & 1 & 3 \\ 0 & 0 & 1 & 3 & 2 & 1 \end{matrix})