1. Linear Equation in Linear Algebra

1. linear equation

1) linear equation in the variables $x1,x2,...,x_n$

• $a_1x_1+a_2x_2+...+a_nx_n=b$
  $a_1,a_2...a_n : coefficients$

2) System of linear equations(linear system)

• a collection of one or more linear equtions

  $x_1 -2x_2= -1$
  $-x_1+3x_2=3$

3) Solution set

• The set of all possible solutions of the linear system
• A linear system 1. no solution,or 2.exactly one solution, or 3. infinitely many solutions
  1 is inconsistent, 2 and 3 are consistent
  

2. Elementary row operations

1) coefficient matrix, augmented matrix

2) replacement, interchange, scaling

3) row equivalent

• two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other
• if the augmented matices of two linear systems are row equivalent, then the two systems have the same solution set

3. Echelon form and Reduced echelon form

1) A leading entry of row

• the leftmost nonzero entry

2) Echelon form
1. All nonzero rows are above any rows of all zeros
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it

3) Reduced echelon form
3. The leading entry in each nonzero row is 1
4. Each leading 1 is the only nonzero entry in its column

Theorem 1. Uniqueness of the Reduced Echelon Form

Each matrix is row equivalent to one and only one reduced echelon matrix

4) Row reduction algorithm

• forward phase > echelon form and backward phase > reduced echelon form

• pivot position in matrix : leading 1 in the reduced echelon form

• general solution : leading position에 해당하는 변수는 basic variables(or leading variables), 나머지 변수를 free variables가 되도록 작성할 것
  자유변수는 어떤한 값을 넣어도 상관없는 변수이며 이에따라 basic variables의 값이 결정된다

Theorem 2. Existence and Uniqueness Theorem

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column

if a linear system is consistent then (i) unique solution(no free variables), or (ii) infinitely solution(at leat one free variables)

3. Vector Equations

1) Linear combinations

• linear combination of $\overrightarrow{v_1},\overrightarrow{v_2},...,\overrightarrow{v_p}$ with weights $c_1,c_2,...,c_p$
  $y=c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+..+c_p\overrightarrow{v_p}$

2) span

• span ${\overrightarrow{v_1},...,\overrightarrow{v_P}}$ is the collection if all vectors that can be written in the form
  $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+...+c_p\overrightarrow{v_p}$

3) 다음은 같은 의미의 question이라고 할 수 있음

• $\begin{bmatrix}\overrightarrow{v_1}&...&\overrightarrow{v_n}&b \end{bmatrix}$ augmented matrix가 해를 갖는가?
  $x_1\overrightarrow{v_1}+x_2\overrightarrow{v_2}+...+x_n\overrightarrow{v_n}=b$가 해를 갖는가?
  $\overrightarrow{b}$가 Span {$\overrightarrow{v_1},\overrightarrow{v_2},...,\overrightarrow{v_n}$}에 있는가