From 9c0c1ba5e2fdcd8bdc19239cb73b145be22834c4 Mon Sep 17 00:00:00 2001 From: eggy Date: Sat, 19 Nov 2022 00:51:18 -0500 Subject: [PATCH] math115: add determinants, adjugates, lintrans --- docs/ce1/math115.md | 152 +++++++++++++++++++++++++++++++++++++++++++- 1 file changed, 151 insertions(+), 1 deletion(-) diff --git a/docs/ce1/math115.md b/docs/ce1/math115.md index b2db6a1..9bff62c 100644 --- a/docs/ce1/math115.md +++ b/docs/ce1/math115.md @@ -748,7 +748,7 @@ Matrix transformation properties: A matrix transformation function can be restored to its original vector by substituting in the standard basis as parameters. -$$A=[f_A(\vec e_1), f_A(\vec e_2), ..., f_A(\vec e_n)]$$ +$$[L]=[L(\vec e_1), L(\vec e_2), ..., L(\vec e_n)]$$ ### Linear transformations @@ -764,3 +764,153 @@ For all linear transformations: - $L(\vec 0_n) = \vec 0_m$ - $L(-\vec x) = -L(\vec x)$ + +Linear combinations **preserve linear combinations**, so a linear transformation of a subspace can be found if the linear transformations of the basis are known. + +### Reflections + +Around a direction vector $\vec d$ through the origin, a linear transformation can reflect about it, similar to a reflection across $y=x$. + +$$ +\begin{align*} +L&: \mathbb R^2 \longrightarrow \mathbb R^2 \\ +&\vec x \longmapsto 2\ \text{proj}_{\vec{d}}\ \vec x-\vec x \\ +&\vec x \longmapsto \vec x - 2\ \text{perp}_{\vec{d}}\ \vec x +\end{align*} +$$ + +In $\mathbb R^3$, it is reflected across a plane, and the normal vector $\vec n$ can be used as the direction vector. + +$$ +\begin{align*} +L&: \mathbb R^3 \longrightarrow \mathbb R^3 \\ +&\vec x\longmapsto \vec x - 2\ \text{proj}_{\vec n}\ \vec x +\end{align*} +$$ + +### Rotations + +Where $R_\theta:\mathbb R^2 \longrightarrow \mathbb R^3$ is a **counterclockwise** rotation about the origin by $\theta$, $r$ is the **norm** of the vector, and $\phi$ is the original angle to the x-axis: + +$$ +R_\theta(\vec x) = \begin{bmatrix} +r\cos(\theta + \phi) \\ +r\sin(\theta + \phi) +\end{bmatrix} +$$ + +This effectively transforms the matrix to: + +$$ +R_\theta(\vec x) = \begin{bmatrix} +\cos\theta & -\sin\theta \\ +\sin\theta & \cos\theta +\end{bmatrix}\vec x +$$ + +The inverse angle is just the matrix transpose. + +### Stretches and compressions + +$$ +Ax=\begin{bmatrix} +t & 0 \\ +0 & 1 +\end{bmatrix} +\begin{bmatrix} +x_1 \\ +x_2 +\end{bmatrix} +$$ + +If $t>1$, $L(\vec x)$ is a **stretch** in the $x_2$ direction. Otherwise, it is a **compression** in that direction. + +$$ +Ax=\begin{bmatrix} +t & 0 \\ +0 & t +\end{bmatrix} +\begin{bmatrix} +x_1 \\ +x_2 +\end{bmatrix} +$$ + +If $t>1$, $L(\vec x)$ is a **dilation** (stretch in both directions). Otherwise, it is a **contraction**. + +$$ +Ax=\begin{bmatrix} +1 & s \\ +0 & 1 +\end{bmatrix} +\begin{bmatrix} +x_1 \\ +x_2 +\end{bmatrix} +$$ + +If $s > 1$, $L(\vec x)$ is a **rightward shear**, bending a square into a right-facing parallelogram. Otherwise, if $0 < s < 1$, it **shears left**. + +For all linear and matrix transformations: + +- $L=M \iff L(\vec x)$ always is equal to $M(\vec x)$ +- $(L+M)\mathbb R^n\to\mathbb R^m: [L] + [M]$ +- $[cL] = c[L]$ + +In $L: \mathbb R^n\to\mathbb R^m, M:\mathbb R^m\to\mathbb R^p$: + +- $[M\circ L]=[M][L]$ or $M(L(\vec x)) = [M][L]\vec x$ + +## Determinants and adjugates + +$$A(\text{adj}\ A)=(\det A)I=(\text{adj}\ A)A$$ + +The determinant does a lot of magic things and is commonly used in the cross product. + +$$ +\det A = \begin{vmatrix} +a & b \\ +c & d +\end{vmatrix} = ad-bc +$$ + +The inverse can be quickly found if and only if the determinant of the original matrix is **non-zero**: + +$$A^{-1}=\frac{1}{\det A}(\text{adj}\ A)$$ + +The mini-matrix $A(i,j)$ is equal to the matrix created by removing the $i$th row and $j$th column. + +The $(i,j)$th cofactor of $A$ is related to its one-indexed determinant, and its sign is determined by its position: + +$$c_{i,j}=\det(A(i,j))(-1)^{i+j}$$ + +The determinant of an arbitrarily sized matrix can be found by recursively applying these formulae — for any row or column, the cofactor expansion along the $i$th row or $j$th column of $A$ can find the determinant: + +$$\det A=a_{i1}c_{i1}+a_{i2}c_{i2} ...$$ + +The **adjugate matrix** is equal to the transpose of the **cofactor matrix**: + +$$\text{adj}\ A=[c_{ij}]^T\text{ for all }i,j$$ + +Determining cofactors is easier with more zeroes in the matrix, so reducing a matrix to RREF via EROs and **switching columns** is simpler. + +- If there is row or column of zeros, $\det A = 0$ +- If $B=A$ with exactly one pair of swapped rows, $\det B = -\det A$ +- If $B=A$ by adding multiples of rows or columns to each other, $\det B = \det A$ +- If $B=A$ with exactly one row or column a scalar multiple $c$ of $A$, $\det B = c\det A$ + +Properties; + +- $\det(AB)=\det(A)\det(B)$ +- $\det(BA)=\det(AB)$ +- $\det(A+B)\neq\det(A)+\det(B)$ +- $\det(cA)=c^n\det(A)$ +- $\det(A^{-1})=\frac{1}{\det A}$ +- $\det A^T = \det A$ + +### Triangular matrices + +A triangular matrix is in REF or REF transposed. + +If $A$ is an upper triangular matrix, it is in REF and all entries below the main diagonal are zero, and $\det A$ is equal to the sum of all entries along the main diagonal. +