"Maymester MATH547 Linear Algebra for Applications in Data Science"

MATH547DS L01: Vectors and Matrices

Overview

Quantities

Numbers in mathematics

$ℕ$: The set of natural numbers, $ℕ = {0, 1, 2, 3, \dots}$ , infinite and countable, $ℕ_{+} = {1, 2, 3, \dots}$ ;
$ℤ$: The set of integers, $ℤ = {0, \pm 1, \pm 2, \pm 3, \dots}$ , infinite and countable;
$ℚ$: The set of rational numbers $ℚ = {p / q, p \in ℤ, q \in ℕ_{+}}$ , infinite and countable;
$ℝ$: The set of real numbers, infinite, not countable, can be ordered;
$ℂ$: The set of complex numbers, $ℂ = {x + i y, x, y \in ℝ}$ , infinite, not countable, cannot be ordered.

Numbers on a computer

Subsets of $ℕ$: The number types uint8, uint16, uint32, uint64 represent subsets of the natural numbers (unsigned integers) using 8, 16, 32, 64 bits respectively.
Subsets of $ℤ$: The number types int8, int16, int32, int64 represent subsets of the integers. One bit is used to store the sign of the number.
Subsets of $ℚ, ℝ, ℂ$: Computers approximate the real numbers through the set $𝔽$ of floating point numbers. Floating point numbers that use $b = 32$ bits are known as single precision, while those that use $b = 64$ are double precision.

Vectors

Table 1. Vector space $𝒱 = (V, S, +, \cdot)$ properties for arbitrary $𝒂, 𝒃, 𝒄 \in V$

Real space, $ℛ_{m} = (ℝ^{m}, ℝ, +, \cdot)$

𝒖 + 𝒗 = [\begin{array}{l} u_{1} \\ ⋮ \\ u_{m} \end{array}] + [\begin{array}{l} v_{1} \\ ⋮ \\ v_{m} \end{array}] = [\begin{array}{l} u_{1} + v_{1} \\ ⋮ \\ u_{m} + v_{m} \end{array}], a 𝒖 = a [\begin{array}{l} u_{1} \\ ⋮ \\ u_{m} \end{array}] = [\begin{array}{l} a u_{1} \\ ⋮ \\ a u_{m} \end{array}] .

(1)

Continuous functions, $𝒞^{0} = (C (ℝ), ℝ, +, \cdot)$ , $(a f) (t) = a f (t)$ , $(f + g) (t) = f (t) + g (t)$ .

Matrices

Matrices are groupings of vectors $𝑨 = [\begin{array}{llll} 𝒂_{1} & 𝒂_{2} & \dots & 𝒂_{n} \end{array}],$ $𝒂_{1}, 𝒂_{2}, \dots, 𝒂_{n} \in V$ , $𝒱 = (V, S, +, \cdot)$
Vectors in $ℛ_{m}$ are given using the identity matrix $𝑰$ , $𝒙 \in ℛ_{m}$ , $𝒙 = 𝑰 𝒙$
$𝒙 = [\begin{array}{l} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{array}] = x_{1} 𝒆_{1} + x_{2} 𝒆_{2} + \dots + x_{m} 𝒆_{m}, 𝒆_{1} = [\begin{array}{l} 1 \\ 0 \\ ⋮ \\ 0 \\ 0 \end{array}], 𝒆_{2} = [\begin{array}{l} 0 \\ 1 \\ ⋮ \\ 0 \\ 0 \end{array}], \dots, 𝒆_{m} = [\begin{array}{l} 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{array}] .$

𝑰 = [\begin{array}{llll} 𝒆_{1} & 𝒆_{2} & \dots & 𝒆_{m} \end{array}] .

Linear combinations

A linear combination is
$𝒃 = x_{1} 𝒂_{1} + x_{2} 𝒂_{2} + \dots x_{n} 𝒂_{n}$
The matrix-vector product is defined to represent linear combinations
$𝑨 = [\begin{array}{llll} 𝒂_{1} & 𝒂_{2} & \dots & 𝒂_{n} \end{array}], 𝒃 = 𝑨 𝒙$

Linear systems

octave]

ex=[1; 0]; ey=[0; 1];

octave]

b=[0.2; 0.4]; I=[ex ey]; I*b

ans = 0.20000 0.40000

octave]

th=pi/6; c=cos(th); s=sin(th);

octave]

tvec=[c; s]; nvec=[-s; c];

octave]

A=[tvec nvec];

octave]

x=A\b

x = 0.37321 0.24641

octave]

[x(1)*tvec x(2)*nvec]

ans = 0.32321 -0.12321 0.18660 0.21340

octave]

Figure 1. Alternative decompositions of force on inclined plane.

Least squares

octave]

m=1000; h=2*pi/m; j=1:m;

octave]

t(j)=(j-1)*h; t=transpose(t);

octave]

n=5; A=[];

octave]

for k=1:n
  A = [A sin(k*t)];
end

octave]

bt=t.*(pi-t).*(2*pi-t);

octave]

x=A\bt;

octave]

b=A*x;

octave]

s=50; i=1:s:m; 
ts=t(i); bs=bt(i);
plot(ts,bs,'ok',t,b,'r');

octave]

print -depsc L01Fig02.eps

octave]

close;

octave]

Figure 2. Comparison of least sqaures approximation (red line) with samples of exact function $b (t)$ .