공학수학2_선형대수

Up: Class_2020_2

1. 2020-10-27 (Linear Algebra)
2. 2020-10-29
3. 가장 중요한 class 하나 놓침
4. 2020-11-05
5. 2020-11-10
6. 2020-11-12
7. 2020-11-17
8. 2020-11-19
9. 2020-11-24
10. 2020-11-26
11. 2020-12-01 Review 1
12. Review 2 놓침

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1. 2020-10-27 (Linear Algebra) ¶

It is abstract. Geometrical representations have limitations.

Vector Space

Example of vector spaces
- $\mathbb{R}^3$
- $\mathbb{R}^n$

그 안에서 덧셈과 스칼라곱이 가능

// 벡터공간,vector_space

Subspace

Example:

원점 (0, 0, 0)을 지나는 평면,plane은 벡터공간 $\mathbb{R}^3$ 의 부분공간이다.

non empty subset that satisfies requirement for vector space
linear combination stay in subspace

"whatever happens in the subspace stays in the subspace."

// 부분공간,subspace

Column space of matrix A

Given matrix A is part of a system of 3 equations and 2 unknowns

Ax=b

$\begin{bmatrix}1&0\\5&4\\2&4\end{bmatrix}\begin{bmatrix}u\\v\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$

only a very thin subset of possible b's with(will인가?) satisfy the equation

$u\begin{bmatrix}1\\5\\2\end{bmatrix}+v\begin{bmatrix}0\\4\\4\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$

(여기서 가장 오른쪽 행렬은)

all possible combination of columns
referred to as column space(a subspace) C(A) of $\mathbb{R}^n$

// 열공간,column_space

Nullspace of matrix A

Nullspace contains all vectors x that gives Ax=0

Example

$\begin{bmatrix}1&0&1\\5&4&9\\2&4&6\end{bmatrix} \begin{bmatrix}c\\c\\-c\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$ ....여기서 두번째 행렬

nullspace is a line

$x=c, y=c, z=-c$

where $c$ is any scalar or number.
즉 영공간은 위 Ax=b에서 x임

// 영공간,null_space

Solving Ax=b

Given Ax=b
여기서

A : 보통 알려져 있음
x : 찾아내려는 것
b : 보통 알려져 있음

(Suppose this describes a physical system that we are trying to find values for some components.)

If $AX_p=b$ and $AX_n=0,$
then $Ax=b$
is $A(X_p+X_n)=b+0$

위에서

$X_p$ : particular solution
$X_n$ : nullspace
$(X_p+X_n)$ : our complete solution

// 해,solution

Example

Given Ax=b as
$\begin{bmatrix}1&3&3&2\\2&6&9&7\\-1&-3&3&4\end{bmatrix}\begin{bmatrix}u\\v\\w\\y\end{bmatrix}=\begin{bmatrix}1\\5\\5\end{bmatrix}$
// 여기서 u,v,w,y를 푸는 것이 목적

We will first re-arrange the above equation into what we called an echelon matrix U.

Forward elimination
// 이건 캡쳐할수밖에...

Approach 1

Approach 2

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2. 2020-10-29 ¶

Linear Independence, Basis and Dimensions

The numbers m(rows) and n(columns) do not give the true size of the linear system.

- it can have zero rows and columns
- combinations of rows of columns

Rank

- gives the true size of the linear system
- is the number of pivots in the elimination process
- genuinely independent rows in matrix A

// 계수,rank

Linear Independence

Given $c_1v_1+c_2v_2+\cdots+c_kv_k=0$

① If the equation can only be satisfied by having $c_1=0,c_2=0,\cdots,c_k=0$ then we say $v_1,v_2,\cdots,v_k$ are linearly independent.

② If any of the coefficients is a non-zero, then we say $v_1,v_2,\cdots,v_k$ are linearly dependent.

// 선형독립,linear_independence

Example

$A=\begin{bmatrix}1&3&3&2\\2&6&9&5\\-1&-3&3&0\end{bmatrix}$
If we perform $c_2=c_2-3c_1$
$A=\begin{bmatrix}1&0&3&2\\2&0&9&5\\-1&0&3&0\end{bmatrix}$ // columns are linearly independent

// 두번째 열이 zero vector

Ax=0
여기서 x: nullspace of A, N(A) must be {zero vectors} if the columns of A are independent.

zero vector를 없애면
$A=\begin{bmatrix}1&3&2\\2&9&5\\-1&3&0\end{bmatrix},$
$\begin{bmatrix}1&3&2\\2&9&5\\-1&3&0\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$
only $x_1=x_2=x_3=0$ satisfy equation

Spanning a subspace

When we say vectors $w_1,w_2,\cdots,w_l$ span the space $V,$ it means vector space $V$ consists of all linear combinations of $w_1,w_2,\cdots,w_l.$
$\operatorname{Space}V=c_1w_1+c_2w_2+\cdots+c_lw_l$
for some coefficient $c_i$
Column space of A is space spanned by its column.
Row space of A is space spanned by its rows.

// 생성,span 부분공간,subspace 열공간,column_space 행공간,row_space

Basis for a vector space

A basis of space $V$ is a set of vectors where

- they are linearly independent (not too many vectors)
- they span the space $V$ (not too few vectors)
Every vector in the space $V$ is a unique combination of basis vectors
If columns of matrix are independent, they are a basis for the column space (and they span it as well)

// 기저,basis

Dimension of a vector space

A space has infinitely many different bases(←plural for basis)
The number of basis vectors is a property of the space

(fixed for a given space V)
number of vectors in the bases = dimension of space
A basis is
- a maximal independent set

- cannot be made larger without losing independence

- cannot be made smaller and still span the space

// 차원,dimension

Four Fundamental Subspaces

Given matrix A is m×n matrix
① Column space of A denoted by C(A) ~ dimension is the rank r
② Null space of A denoted by N(A) ~ dimension is n-r
③ Row space of A is the column space of A^T ~ dimension is r
④ Left nullspace of A is the nullspace of A^T ~ dimension is m-r

It contains all vectors y such that A^Ty=0 denoted by N(A^T)

Q&A left nullspace란?
2번에서 Ax=0에서 x nullspace가 맞는지 CHK
4번에서 y가 left nullspace이다.

① Column space of A

Pivot columns of A are a basis for its column space
If sets of columns in A are independent, it corresponding columns in echelon matrix V are also independent.

Example

$V=\begin{bmatrix}d_1&*&*&*&*&0\\0&0&0&d_2&*&0\\0&0&0&0&0&d_3\\0&0&0&0&0&0\end{bmatrix}$

Assumed columns 1,4,6 are independent columns
Columns 1,4,6 are basis for C(A)
Row rank = column rank (important theorem in linear algebra)
If rows of square matrix are independent, the columns are also independent

// 질문생략

Question

Is dimension of subspace made by 2 vectors (1 2 1)^T and (1 0 0)^T two? Even if the number of variables is three and the plane is on vector space dimension 3?

// 질문생략

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3. 가장 중요한 class 하나 놓침 ¶

circuit network matrix

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4. 2020-11-05 ¶

Nullspace of A

$Ax=0,\;x=\begin{bmatrix}c\\c\\c\\c\end{bmatrix}$

Column space of A

Is b in the column space of A (what values of b will satisfy Ax=b)
....포기, 캡쳐 참조

// 열공간,column_space

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5. 2020-11-10 ¶

Linear Transformations

// 선형변환,linear_transformation

Transformations represented by matrices

The same transformation can be described by another set of basis vectors

Example
// basis가 다항식,polynomial... 행렬로 하는 미적분? $A_{\textrm{diff}}$

And we can do the same for integration - integration matrix $A_{\textrm{int}}$

Transformation of the plane
Stretching
Rotation by 90° (ccw)

Reflection (45° line)
Projection (onto x axis) // 사영,projection

Rotation through angle θ

// 회전,rotation

Projection onto θ-line
(θ-line은 x축에서 반시계방향으로 회전한 선)

// 사영,projection

Reflection about mirror θ-line

// 반사,reflection

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6. 2020-11-12 ¶

Orthogonal vectors and subspaces

Length of vector

Given vector $\vec{x}=( x_1,x_2,\cdots,x_n )$
Length squared is $||\vec{x}||^2=x_1^2+x_2^2+\cdots+x_n^2$

$=\vec{x}{}^T\vec{x}$

Example

Given $\vec{x}=\begin{bmatrix}1\\2\\-3\end{bmatrix}$
length square of $\vec{x}$ is $\vec{x}{}^T\vec{x}=[1\;2\;3]\begin{bmatrix}1\\2\\-3\end{bmatrix}=14$

// x^T는 x의 전치(see 전치행렬,transpose_matrix)