# Linear Algebra/Notation

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### Notation

 $\mathbb{R}$, $\mathbb{R}^+$, $\mathbb{R}^n$ set of real numbers, reals greater than $0$, ordered $n$-tuples of reals $\mathbb{N}$ natural numbers: $\{0,1,2,\ldots\}$ $\mathbb{C}$ complex numbers $\{\ldots\,\big|\,\ldots\}$ set of . . . such that . . . $(a\,..\,b)$, $[a\,..\,b]$ interval (open or closed) of reals between $a$ and $b$ $\langle \ldots \rangle$ sequence; like a set but order matters $V,W,U$ vector spaces $\vec{v},\vec{w}$ vectors $\vec{0}$, $\vec{0}_V$ zero vector, zero vector of $V$ $B,D$ bases $\mathcal{E}_n=\langle \vec{e}_1,\,\ldots,\,\vec{e}_n \rangle$ standard basis for $\mathbb{R}^n$ $\vec{\beta},\vec{\delta}$ basis vectors ${\rm Rep}_{B}(\vec{v})$ matrix representing the vector $\mathcal{P}_n$ set of $n$-th degree polynomials $\mathcal{M}_{n \! \times \! m}$ set of $n \! \times \! m$ matrices $[S]$ span of the set $S$ $M\oplus N$ direct sum of subspaces $V\cong W$ isomorphic spaces $h,g$ homomorphisms, linear maps $H,G$ matrices $t,s$ transformations; maps from a space to itself $T,S$ square matrices ${\rm Rep}_{B,D}(h)$ matrix representing the map $h$ $h_{i,j}$ matrix entry from row $i$, column $j$ $\left|T\right|$ determinant of the matrix $T$ $\mathcal{R}(h),\mathcal{N}(h)$ rangespace and nullspace of the map $h$ $\mathcal{R}_\infty(h),\mathcal{N}_\infty(h)$ generalized rangespace and nullspace

### Lower case Greek alphabet

$\begin{array}{ll|ll|ll} \text{name} &\text{character} &\text{name} &\text{character} &\text{name} &\text{character} \\ \hline \text{alpha} & \alpha &\text{iota} & \iota &\text{rho} & \rho \\ \text{beta} & \beta &\text{kappa} & \kappa &\text{sigma} & \sigma \\ \text{gamma} & \gamma &\text{lambda} & \lambda &\text{tau} & \tau \\ \text{delta} & \delta &\text{mu} & \mu &\text{upsilon}& \upsilon \\ \text{epsilon} & \epsilon &\text{nu} & \nu &\text{phi} & \phi \\ \text{zeta} & \zeta &\text{xi} & \xi &\text{chi} & \chi \\ \text{eta} & \eta &\text{omicron}& o &\text{psi} & \psi \\ \text{theta} & \theta &\text{pi} & \pi &\text{omega} & \omega \end{array}$

About the Cover. This is Cramer's Rule for the system $x_1+2x_2=6$, $3x_1+x_2=8$. The size of the first box is the determinant shown (the absolute value of the size is the area). The size of the second box is $x_1$ times that, and equals the size of the final box. Hence, $x_1$ is the final determinant divided by the first determinant.

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