Title: Theory / Important / Math / Analysis / Vector Calculus, Linear Algebra and Differential Forms+17
Tags: fromMath mid1516974248815 toUnary

Definition: \(\sum_{(m_1,\dots,m_n)\in\text{perm }n}\left(\text{sign}(m_1,\dots,m_n)\right)A_{m_1,1}\dots A_{m_n,n}.\)
Page: 314, 317, 75
Section: B,, 4
Chapter: 10,, 1
Variables: \(\mathbb F\) a field
\(A\) be an \(n\)-by-\(n\) matrix.
Name: The determinant of \(A\)
Index: Definition
Kind: 33 .37, 39
Linear: Linear in each row/column
Abbreviation: det \(A\)
Definition2: the unique function \(f\), linear on each column, such that \(f(I)=1\) and such that permuting column change the sign of the determinant.
Assuming: Nothing
Intuition: The signed measure of the parallelogram with all column vectors of \(A\)

Name: The angle between \(\vec v\) and \(\vec w\)
Definition: \(\arccos\left(\frac{\vec v\cdot\vec w}{|\vec v||\vec w|}\right)\)
Page: 71
Chapter: 1
Section: 4
Variables: \(n\in\mathbb N\)
\(\vec v, \vec w\in\mathbb R^n\)
Index: 6
Kind: Def

Name: The length of \(A\)
Definition: \[\sqrt{\sum_{1\le i\le n, 1\le j\le m}a_{i,j}^2}\]
Page: 73
Chapter: 1
Section: 4
Variables: \(n,m\in\mathbb N\)
\(A\in\mathbb R^{n,m}\)
Notation: \(|A|\)
Index: 10
Name2: Probenius norm of \(A\)
Name3: The Schur norm of \(A\)
Name4: The Hilbert-Schmidt norm of \(A\)

Smaller0: \(|A\vec b|\)
Greater0: \(|A||\vec b|\)
Page: 73
Chapter: 1
Section: 4
Variables: \(n,m\in\mathbb N\)
\(A\in\mathbb R^{n,m}\)
\(\vec a\in\mathbb R^m\)
Index: 11
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(|AB|\)
Greater0: \(|A||B|\)
Page: 73
Chapter: 1
Section: 4
Variables: \(n,m\in\mathbb N\)
\(A\in\mathbb R^{n,m}\)
\(B\in\mathbb R^{m, k}\)
Index: 11
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(\lim_{\epsilon\to0}u(\epsilon)=0\)
Greater0: you can use \(u(\epsilon)\) instead of \(\epsilon\) to define/prove convergence
Page: 89
Chapter: 1
Section: 5
Variables: \(u:\mathbb R\to\mathbb R\)
Index: 14
Kind: Prop

Smaller0: \(\lim_{i\to\infty}(\vec a_i)_{i\in\mathbb N},\lim_{i\to\infty}(\vec b_i)_{i\in\mathbb N}\) both converges
Greater0: \(\lim_{i\to\infty}\vec a_i\vec b_i\)
Page: 89
Chapter: 1
Section: 5
Variables: \(m\in\mathbb N\)
\((\vec a_i)_{i\in\mathbb N},(\vec b_i)_{i\in\mathbb N}\in\mathbb R^m\)
Index: 14
Kind: Prop
Greater1: \(\lim_{i\to\infty}\vec a_i\lim_{i\to\infty}\vec b_i\)
Connect Greater0: =

Smaller0: \(\lim_{i\to\infty}(\vec a_i)_{i\in\mathbb N},\lim_{i\to\infty}(\vec b_i)_{i\in\mathbb N}\) both converges
Greater0: \(\lim_{i\to\infty}\vec a_i\cdot\vec b_i\)
Page: 89
Chapter: 1
Section: 5
Variables: \(m\in\mathbb N\)
\((\vec a_i)_{i\in\mathbb N},(\vec b_i)_{i\in\mathbb N}\in\mathbb R^m\)
Index: 14
Kind: Prop
Greater1: \(\lim_{i\to\infty}\vec a_i\cdot\lim_{i\to\infty}\vec b_i\)
Connect Greater0: =

Smaller0: \(\lim_{i\to\infty}(\vec a_i)_{i\in\mathbb N},\lim_{i\to\infty}(c_i)_{i\in\mathbb N}\) both converges
Greater0: \(\lim_{i\to\infty}\vec a_i\cdot c_i\)
Page: 89
Chapter: 1
Section: 5
Variables: \(m\in\mathbb N\)
\((c_i)_{i\in\mathbb N}\in\mathbb R\)
\((\vec a_i)_{i\in\mathbb N}\in\mathbb R^m\)
Index: 14
Kind: Prop
Greater1: \(\lim_{i\to\infty}\vec a_i\cdot\lim_{i\to\infty}c_i\)
Connect Greater0: =

Smaller0: \((I-A)^{-1}\)
Greater0: \(\sum_{i=0}^{\infty}A^i\)
Page: 100
Chapter: 1
Section: 5
Variables: \(m\in\mathbb N\)
\(A\in\mathbb R^{m,m}\)
Index: 14
Kind: Prop
Smaller to greater: =
Assuming: \(|A|<1\)

Smaller0: \(|A|<1\)
Greater0: \((I-A)^{-1}\) is invertible, with inverse \(\sum_{i=0}^{\infty}A^i\)
Page: 101
Chapter: 1
Section: 5
Variables: \(m\in\mathbb N\)
\(A\in\mathbb R^{m,m}\)
Index: 39
Kind: Prop

Question: The set of invertible matrices is
Answer: open
Page: 101
Chapter: 1
Section: 5
Index: 39
Kind: Prop