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Introduction to Linear Algebra with Mathematica

Glossary

Notations

List of Symbols

๐”ฝ       field (usualy either โ„š or โ„ or โ„‚)
โ„       field of real numbers
โ„š       field of rational numbers
โ„‚       field of complex numbers
j       imaginary unit (the vector in the positive vertical direction on complex plane โ„‚), so jยฒ = โˆ’1
z*       complex conjugate:   z* = (๐‘Ž + jโ€‰b)* = ๐‘Ž โˆ’ jโ€‰b
ยฏz       complex conjugate:   ยฏz=ยฏ(a+jb)=aโˆ’jb
A*       adjoint matrix: ยฏAT, transpose and complex conjugate.
AT       transpose matrix is obtained by changing the rows into columns.
โ„•       the set nonnegative integers: 0, 1, 2, โ€ฆ
โ„ค       the set of all integers: 0, ยฑ1, ยฑ2, ยฑ3, โ€ฆ
๐•‹       โ„/(2ฯ€โ„ค) unit circle or one-dimensional tores.
O(g(n))       big-oh is also called Bachmannโ€“Landau notation or asymptotic notation: |f(n)|โ‰คM|g(n)| as n โ†’ โˆž.
o(g(n))       little=oh means lim
n!       factorial:    1ยท2ยท3ยท โ‹ฏ ยทn
\displaystyle n^{\underline{m}}       falling factorial   \displaystyle n^{\underline{m}} = n\left( n-1 \right)\left( n-2 \right) \cdots \left( n-m+1 \right)
\displaystyle n^{\overline{m}}       rising factorial (or Pochhammer symbol)   \displaystyle n^{\overline{m}} = n \left( n+1 \right)\left( n+2 \right) \cdots \left( n+m-1 \right)
(2n)!!       double factorial:   (2n)!! = (2n) ยท โ‹ฏ ยท 2 (2nโˆ’2) ยท (2nโˆ’4) ยท
(2n+1)!!       double factorial:   (2n+1)!! = (2n+1) ยท (2nโˆ’1) ยท (2nโˆ’3) ยท โ‹ฏ ยท 1
\displaystyle \binom{n}{k}       binomial coefficient:   \displaystyle \binom{n}{k} = \frac{n^{\underline{k}}}{k!} , where k โˆˆ โ„•
(๐‘Ž, b)       open interval on โ„ (๐‘Ž or b or both can be infinity)
[๐‘Ž, b]       closed interval
|๐‘Ž, b|       any interval with endpoints |๐‘Ž and b; it can be closed, open, or semi-closed
AโˆฉB       intersection of two sets
AโˆชB       union of two sets
\displaystyle \overline{\Omega}       closure of set ฮฉ
โˆ‚ฮฉ       boundary of set ฮฉ
โ‡€       weak convergence: fn โ‡€ f iff โŸจ u | fn โŸฉ โ†’ โŸจ u | f โŸฉ for any u โˆˆ โ„Œ
     
     
     

Vector Spaces

๐”ฝn       direct product of n fields ๐”ฝร—๐”ฝร— โ‹ฏ; ร—๐”ฝ.
โ„n       real Cartesian product
โ„‚n       complex Cartesian product
๐”ฝm,n       set of all mโ€‰ร—โ€‰n matrices
๐”ฝ[x]       Set of polynomials of variable x over field ๐”ฝ, also denoted by โ„˜, โ„˜
๐”ฝโ‰คn[x]       Set of polynomials over field ๐”ฝ of degree less than or equal to n.
๐’ž(A)       Column space of matrix A D49E;
โ„›(A)       Row space of matrix A.
๐’ฉ(A)       Null space of matrix A, also known as the kernel of matrix A.
ker(A)       Kernel or Null space of matrix A, so ker(A) = ๐’ฉ(A).
coker(A)       Cokernel of matrix A is the kernel of adjoint matrix A*.
โŸจ f , g โŸฉ       inner product (in mathematics)
โŸจ f | g โŸฉ       inner product (in physics)
โˆฅยทโˆฅ       norm in a normed space
โ„“ยฒ       or โ„“2 is the set of sequence with norm \displaystyle \| {\bf x} \|_2 = \left( \sum_{i\ge 0} |x_i |^2 \right)^{1/2}
๐”ยฒ[๐‘Ž, b]       set of square integrable (Lebesgue) functions on the interval [๐‘Ž, b]
๐”ยฒ([๐‘Ž, b], w)       set of square integrable (Lebesgue) functions with weight w on the interval [๐‘Ž, b]
โ„ญ[๐‘Ž, b]       set of continuous functions on interval [๐‘Ž, b]
โ„ญm[ฮฉ]       m-times continuously differentiable functions ฮฉ โ†’ โ„‚
๐’ฎ(โ„)       Schwartz functions (smooth functions with rapid decay), also denoted by S(โ„)
๐’ฎ*(โ„)       set of tempered distributions, also denoted by ๐’ฎ'(โ„) or S'(โ„)
     
     

Operators

\displaystyle \texttt{D}       differential or derivative operator in Euler's notation:   \displaystyle \texttt{D} = \frac{\text d}{{\text d}x} with respect to variable x
\displaystyle \frac{{\text d}y}{{\text d}x}       derivative of function y in Leibniz's notation
y'       derivative of function y in Lagrange's notation
\displaystyle \dot{y}       derivative of function y in Newton's notation with respect to time variable: \displaystyle \dot{y} = {\text d}y/{\text d}t
โˆ‚       partial derivative
โˆ‚xu       partial derivative of u with respect to variable x, also denoted as ux or \displaystyle \frac{\partial u}{\partial x}
\displaystyle \hat{p}       momentum operator:   \displaystyle \hat{p} = -{\bf j}\,\hbar\,\partial , where ฤง is Planck's reduced constant
โˆ‡       gradient operator
ฮ”       Laplace operator \displaystyle \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}
โ–ก       d'Alembert operator \displaystyle \square = \frac{\partial^2}{\partial t^2} - c^2 \nabla^2 = \frac{\partial^2}{\partial t^2} - c^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right)
S[f]       (formal) Fourier series of function f in either exponential form or trigonometric form
S*[f]       conjugate Fourier series
SN(f; x)       N-th partial Fourier sum \displaystyle \sum_{n=-N}^N \hat{f}(n) \,e^{{\bf j} n\pi x/\ell} = \frac{a_0}{2} + \sum_{k=1}^N a_k \cos \frac{k\pi x}{\ell} + b_k \sin \frac{k\pi x}{\ell}
Iโ€‰fโ€‰โ‰ซ       list of Fourier coefficients either in complex or trigonometric form
โ„ฑ\left[ f \right]       Fourier transform \displaystyle {\hat {f}} or โ„ฑ\left[ f \right] or f^F .
โ„ฑ^{-1}\left[ f^F \right]       inverse Fourier transform
fโ˜…g       convolution: f\star g (x) = \int f(y)\,g(x-y)\,{\text d} y
โ„’[ f ]       Laplace transform \displaystyle f^L (\lambda ) or โ„’\left[ f \right] or f^L = \int_0^{\infty} f(t)\,e^{-\lambda t}{\text d} t
โ„’โˆ’1[ fL ]       inverse Laplace transform \displaystyle โ„’^{-1}\left[ f^L \right] = \mbox{P.V.} \frac{1}{2\pi{\bf j}} \int_{h-{\bf j}\infty}^{h+{\bf j}\infty} f^L (\lambda )\,e^{\lambda\,t} {\text d}\lambda
     
     

Functions

lnx       natural logarithm with base e
ฮ“(ฮฝ)       gamma function \displaystyle \Gamma (\nu ) = \int_0^{\infty} t^{\nu -1} e^{-t} {\text d} t
ฯ‡A       characteristic (or indicator) function of a set A
ฮด(x)       delta function of Dirac
H(t)       Heaviside function:   \displaystyle H(t) = \begin{cases} 1, & \quad t> 0, \\ ยฝ , & \quad t = 0, \\ 0, & \quad t < 0 . \end{cases}