# Linear Algebra/Notation

 Linear Algebra ← Cover Notation Introduction →

## Notation

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

### Lower case Greek alphabet

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

About the Cover. This is Cramer's Rule for the system ${\displaystyle x_{1}+2x_{2}=6}$ , ${\displaystyle 3x_{1}+x_{2}=8}$ . 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 ${\displaystyle x_{1}}$  times that, and equals the size of the final box. Hence, ${\displaystyle x_{1}}$  is the final determinant divided by the first determinant.

 Linear Algebra ← Cover Notation Introduction →