Introduction to Linear Algebra with Mathematica

# Notations

## List of Symbols

 𝔽 field (usualy either ℚ or ℝ or ℂ) ℝ field of real numbers ℚ field of rational numbers ℂ field of complex numbers j imaginary unit (the vector in the positive vertical direction on complex plane ℂ), so j² = −1 z* complex conjugate:   z* = (𝑎 + j b)* = 𝑎 − j b $$\displaystyle \overline{z}$$ complex conjugate:   $$\displaystyle \overline{z} = \overline{(a + {\bf j}\, b)} = a - {\bf j}\, b$$ A* adjoint matrix: $$\displaystyle \overline{{\bf A}^{\mathrm{T}}} ,$$ transpose and complex conjugate. V ≌ W isomorphism between V and W. AT transpose matrix is obtained by changing the rows into columns. ℕ the set nonnegative integers: 0, 1, 2, … ℤ the set of all integers: 0, ±1, ±2, ±3, … 𝕋 ℝ/(2πℤ) unit circle or one-dimensional tores. O(g(n)) big-oh is also called Bachmann–Landau notation or asymptotic notation: $$\displaystyle \left\vert f(n) \right\vert \le M\,\left\vert g(n) \right\vert$$ as n → ∞. o(g(n)) little=oh means $$\displaystyle \lim_{n\to\infty} \frac{f(n)}{g(n)} = 0 .$$ n! factorial:    1·2·3· ⋯ ·n $$\displaystyle n^{\underline{m}}$$ 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) · ⋯ · 2 (2n−2) · (2n−4) · (2n+1)!! double factorial:   (2n+1)!! = (2n+1) · (2n−1) · (2n−3) · ⋯ · 1 $$\displaystyle \binom{n}{k}$$ binomial coefficient:  $$\displaystyle \binom{n}{k} = \frac{n^{\underline{k}}}{k!} ,$$ where k ∈ ℕ (𝑎, b) open interval on ℝ (𝑎 or b or both can be infinity) [𝑎, b] closed interval |𝑎, b| any interval with endpoints |𝑎 and b; it can be closed, open, or semi-closed A∩B intersection of two sets A∪B union of two sets $$\displaystyle \overline{\Omega}$$ closure of set Ω ∂Ω boundary of set Ω ⇀ weak convergence: fn ⇀ f iff ⟨ u | fn ⟩ → ⟨ u | f ⟩ for any u ∈ ℌ

## Vector Spaces

 𝔽n direct product of n fields 𝔽×𝔽× ⋯; ×𝔽. ℝn real Cartesian product ℂn complex Cartesian product 𝔽m,n set of all m × n matrices 𝔽[x] Set of polynomials of variable x over field 𝔽, also denoted by ℘, ℘ 𝔽≤n[x] Set of polynomials over field 𝔽 of degree less than or equal to n. 𝒞(A) Column space of matrix A D49E; ℛ(A) Row space of matrix A. 𝒩(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) = 𝒩(A). coker(A) Cokernel of matrix A is the kernel of adjoint matrix A*. ⟨ f , g ⟩ inner product (in mathematics) ⟨ f | g ⟩ inner product (in physics) ∥·∥ norm in a normed space ℓ² or ℓ2 is the set of sequence with norm $$\displaystyle \| {\bf x} \|_2 = \left( \sum_{i\ge 0} |x_i |^2 \right)^{1/2}$$ 𝔏²[𝑎, b] set of square integrable (Lebesgue) functions on the interval [𝑎, b] 𝔏²([𝑎, b], w) set of square integrable (Lebesgue) functions with weight w on the interval [𝑎, b] ℭ[𝑎, b] set of continuous functions on interval [𝑎, b] ℭm[Ω] m-times continuously differentiable functions Ω → ℂ 𝒮(ℝ) Schwartz functions (smooth functions with rapid decay), also denoted by S(ℝ) 𝒮*(ℝ) set of tempered distributions, also denoted by 𝒮'(ℝ) or S'(ℝ)

## 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: $$\displaystyle \dot{y} = {\text d}y/{\text d}t$$ ∂ partial derivative ∂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 ħ is Planck's reduced constant ∇ gradient operator Δ Laplace operator $$\displaystyle \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$$ □ 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 $$\displaystyle \sum_{n=-N}^N \hat{f}(n) \,e^{{\bf j} n\pi x/\ell} = \frac{a_0}{2} + \sum_{k=1}^N a_k \cos \frac{k\pi x}{\ell} + b_k \sin \frac{k\pi x}{\ell}$$ I f ≫ list of Fourier coefficients either in complex or trigonometric form $$ℱ\left[ f \right]$$ Fourier transform $$\displaystyle {\hat {f}}$$ or $$ℱ\left[ f \right]$$ or $$f^F .$$ $$ℱ^{-1}\left[ f^F \right]$$ inverse Fourier transform f★g convolution: $$f\star g (x) = \int f(y)\,g(x-y)\,{\text d} y$$ ℒ[ f ] Laplace transform $$\displaystyle f^L (\lambda )$$ or $$ℒ\left[ f \right]$$ or $$f^L = \int_0^{\infty} f(t)\,e^{-\lambda t}{\text d} t$$ ℒ−1[ fL ] inverse Laplace transform $$\displaystyle ℒ^{-1}\left[ f^L \right] = \mbox{P.V.} \frac{1}{2\pi{\bf j}} \int_{h-{\bf j}\infty}^{h+{\bf j}\infty} f^L (\lambda )\,e^{\lambda\,t} {\text d}\lambda$$

## Functions

 lnx natural logarithm with base e Γ(ν) gamma function $$\displaystyle \Gamma (\nu ) = \int_0^{\infty} t^{\nu -1} e^{-t} {\text d} t$$ χA characteristic (or indicator) function of a set A δ(x) delta function of Dirac H(t) Heaviside function:   $$\displaystyle H(t) = \begin{cases} 1, & \quad t> 0, \\ ½ , & \quad t = 0, \\ 0, & \quad t < 0 . \end{cases}$$