221 lines
5.5 KiB
TeX
221 lines
5.5 KiB
TeX
\chapter{Elements of Linear Algebra}
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\label{ch:elements-of-linear-algebra}
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\begin{remark}[vector]
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Let $u$ a vector, we will use interchangeably the following notations: $u$ and $\vec{u}$
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\end{remark}
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Let $u = \begin{pmatrix}
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u_1 \\
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\vdots \\
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u_n
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\end{pmatrix}$ and $v = \begin{pmatrix}
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v_1 \\
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\vdots \\
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v_n
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\end{pmatrix}$
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\begin{definition}[Scalar Product (Dot Product)]
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\begin{align*}
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\scalar{u, v} & = \begin{pmatrix}
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u_1, \ldots, u_v
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\end{pmatrix}
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\begin{pmatrix}
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v_1 \\
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\vdots \\
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v_n
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\end{pmatrix} \\
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& = u_1 v_1 + u_2 v_2 + \ldots + u_n v_n
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\end{align*}
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We may use $\scalar{u, v}$ or $u \cdot v$ notations.
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\end{definition}
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\paragraph{Dot product properties}
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\begin{description}
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\item[Commutative] $\scalar{u, v} = \scalar{v, u}$
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\item[Distributive] $\scalar{(u+v), w} = \scalar{u, w} + \scalar{v, w}$
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\item $\scalar{u, v} = \norm{u} \times \norm{v} \times \cos(\widehat{u, v})$
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\item $\scalar{a, a} = \norm{a}^2$
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\end{description}
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\begin{definition}[Norm]
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Length of the vector.
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\[
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\norm{u} = \sqrt{\scalar{u, v}}
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\]
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$\norm{u, v} > 0$
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\end{definition}
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\begin{definition}[Distance]
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\[
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dist(u, v) = \norm{u-v}
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\]
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\end{definition}
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\begin{definition}[Orthogonality]
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\end{definition}
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\begin{remark}
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\[
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(dist(u, v))^2 = \norm{u - v}^2,
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\] and
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\[
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\scalar{v-u, v-u}
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\]
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\end{remark}
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\begin{figure}
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\centering
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\includestandalone{figures/schemes/vector_orthogonality}
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\caption{Scalar product of two orthogonal vectors.}
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\label{fig:scheme-orthogonal-scalar-product}
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\end{figure}
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\begin{align*}
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\scalar{v-u, v-u} & = \scalar{v, v} + \scalar{u, u} - 2 \scalar{u, v} \\
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& = \norm{v}^2 + \norm{u}^2 \\
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& = -2 \scalar{u, v}
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\end{align*}
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\begin{align*}
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\norm{u - v}^2 & = \norm{u}^2 + \norm{v}^2 - 2 \scalar{u,v} \\
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\norm{u + v}^2 & = \norm{u}^2 + \norm{v}^2 + 2 \scalar{u,v}
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\end{align*}
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\begin{proposition}[Scalar product of orthogonal vectors]
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\[
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u \perp v \Leftrightarrow \scalar{u, v} = 0
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\]
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\end{proposition}
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\begin{proof}[Indeed]
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$\norm{u-v}^2 = \norm{u+v}^2$, as illustrated in \autoref{fig:scheme-orthogonal-scalar-product}.
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\begin{align*}
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\Leftrightarrow & -2 \scalar{u, v} = 2 \scalar{u, v} \\
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\Leftrightarrow & 4 \scalar{u, v} = 0 \\
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\Leftrightarrow & \scalar{u, v} = 0
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\end{align*}
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\end{proof}
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\begin{theorem}[Pythagorean theorem]
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If $u \perp v$, then $\norm{u+v}^2 = \norm{u}^2 + \norm{v}^2$ .
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\end{theorem}
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\begin{definition}[Orthogonal Projection]
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\end{definition}
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Let $y = \begin{pmatrix}
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y_1 \\
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. \\
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y_n
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\end{pmatrix} \in \RR[n]$ and $w$ a subspace of $\RR[n]$.
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$\mathcal{Y}$ can be written as the orthogonal projection of $y$ on $w$:
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\[
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\mathcal{Y} = proj^w(y) + z,
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\]
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where
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\[
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\begin{cases}
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z \in w^\perp \\
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proj^w(y) \in w
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\end{cases}
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\]
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There is only one vector $\mathcal{Y}$ that ?
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The scalar product between $z$ and (?) is zero.
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\begin{property}
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$proj^w(y)$ is the closest vector to $y$ that belongs to $w$.
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\end{property}
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\begin{definition}[Matrix]
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A matrix is an application, that is, a function that transform a thing into another, it is a linear function.
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\end{definition}
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\begin{example}[Matrix application]
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Let $A$ be a matrix:
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\[
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A = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}
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\] and
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\[
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x = \begin{pmatrix}
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x_1 \\
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x_2
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\end{pmatrix}
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\]
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Then,
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\begin{align*}
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Ax & = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}
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\begin{pmatrix}
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x_1 \\
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x_2
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\end{pmatrix} \\
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& = \begin{pmatrix}
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a x_1 + b x_2 \\
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c x_1 + d x_2
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\end{pmatrix}
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\end{align*}
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Similarly,
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\begin{align*}
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\begin{pmatrix}
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a & b & c & d \\
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e & f & g & h \\
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i & j & k & l
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\end{pmatrix}
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\begin{pmatrix}
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x_1 \\
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x_2 \\
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x_3 \\
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x_4
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\end{pmatrix}
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=
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\begin{pmatrix}
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\luadirect{
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local matrix_product = require("scripts.matrix_product")
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local m1 = {
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{"a", "b", "c", "d"},
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{"e", "f", "g", "h"},
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{"i", "j", "k", "l"}
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}
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local m2 = {
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{"x_1"},
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{"x_2"},
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{"x_3"},
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{"x_4"}
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}
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local product_matrix = matrix_product.matrix_product_repr(m1,m2)
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local matrix_dump = matrix_product.dump_matrix(product_matrix)
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tex.print(matrix_dump)
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}
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\end{pmatrix}
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\end{align*}
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\end{example}
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The number of columns has to be the same as the dimension of the vector to which the matrix is applied.
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\begin{definition}[Tranpose of a Matrix]
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Let $A = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}$, then $A^T = \begin{pmatrix}
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a & c \\
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b & d
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\end{pmatrix}$
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\end{definition}
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\begin{figure}
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\centering
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\includestandalone{figures/schemes/coordinates_systems}
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\caption{Coordinate systems}
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\end{figure}
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