# Linear Algebra/Notation

 Linear Algebra ← Cover Notation Introduction →

## 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.

 Linear Algebra ← Cover Notation Introduction →