Linear Algebra/Notation

Notation

$\mathbb {R}$ , $\mathbb {R} ^{+}$ , $\mathbb {R} ^{n}$	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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