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Mathematical expressions
1. Vectors $\mathbf{u}$, $\mathbf{a}$ and $\mathbf{b}$:

$\mathbf{u}= \alpha \mathbf{a}+ \beta \mathbf{b}\phantom{\rule{2em}{0ex}}$ Norm:$||\mathbf{u}||$.

$\begin{array}{lll}\hfill \stackrel{}{\mathbf{u}}& = \lambda \mathbf{a}=\frac{〈\mathbf{a},\mathbf{u}〉}{〈\mathbf{a},\mathbf{a}〉}\mathbf{a}=\frac{\left(\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill -2\hfill \\ \hfill -1\hfill \end{array}\right)}{\left(\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \end{array}\right)}\mathbf{a}=\frac{5}{10}\mathbf{a}=\frac{1}{2}\mathbf{a}=\left(\begin{array}{c}\hfill -1.5\hfill \\ \hfill 0.5\hfill \end{array}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\end{array}$
2. Matrices $\begin{array}{llllllllllll}\hfill \left(\begin{array}{cc}\hfill 1\hfill & \hfill 3 + 2i\hfill \\ \hfill 3 - 2i\hfill & \hfill 4\hfill \end{array}\right)& \phantom{\rule{2em}{0ex}}& \hfill det\mathbf{A}& =\left|\begin{array}{ccc}\hfill 3\hfill & \hfill 4\hfill & \hfill 7\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 5\hfill \end{array}\right|\phantom{\rule{2em}{0ex}}& \hfill \left|\begin{array}{ccc}\hfill x\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill x\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill x\hfill \end{array}\right|& ={\left(x - 1\right)}^{2}\left(x + 2\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
3. ${a}_{1}\ge 0\phantom{\rule{2em}{0ex}}p\left(x\right) ={p}_{0}+{p}_{1}x +{p}_{2}{x}^{2}+{p}_{3}{x}^{3}+{p}_{4}{x}^{4}+{p}_{5}{x}^{5}$
4. Transformations: