Linear Algebra/Notation

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Linear Algebra
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Notation[edit]

 \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[edit]


\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
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