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Mathematical expressions

Vectors $u$ , $a$ and $b$ : $\begin{align} u & = (\begin{matrix} −2 \\ −1 \end{matrix}) & a & = (\begin{matrix} −3 \\ 1 \end{matrix}) & b & = (\begin{matrix} 1 \\ 3 \end{matrix}) & (1) \end{align}$
$u = α a + β b$ Norm: $|| u ||$ .
$\begin{align} \hat{u} & = λ a = \frac{〈 a, u 〉}{〈 a, a 〉} a = \frac{(\begin{matrix} −3 \\ 1 \end{matrix}) ⋅ (\begin{matrix} −2 \\ −1 \end{matrix})}{(\begin{matrix} −3 \\ 1 \end{matrix}) ⋅ (\begin{matrix} −3 \\ 1 \end{matrix})} a = \frac{5}{10} a = \frac{1}{2} a = (\begin{matrix} −1.5 \\ 0.5 \end{matrix}) & (2) \end{align}$
Matrices $\begin{array}{l} (\begin{matrix} 1 & 3 + 2i \\ 3 − 2i & 4 \end{matrix}) & det A & = |\begin{matrix} 3 & 4 & 7 \\ 2 & −2 & 1 \\ 1 & 2 & 5 \end{matrix}| & |\begin{matrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{matrix}| & = {(x − 1)}^{2} (x + 2) \end{array}$
$a_{1} ≥ 0 p(x) = p_{0} + p_{1} x + p_{2} x^{2} + p_{3} x^{3} + p_{4} x^{4} + p_{5} x^{5}$
Transformations: $\begin{array}{l} T : ℝ^{3} & → ℝ^{2} & where T (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) = (\begin{matrix} a_{1} a_{2} \\ a_{1} a_{3} \end{matrix}) \end{array}$

$\begin{array}{l} T : P_{3} & → M_{22} & where T (\begin{matrix} a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} \end{matrix}) = (\begin{matrix} a_{1} & a_{2} \\ a_{3} & a_{1} − a_{0} \end{matrix}) \end{array}$