『大学学习』LA乱记

随便写写

Chapter 1

Section 1.1: Systems of Linear Equations

a over b means \(\large\frac ab\)

linear system = a system of linear equations线性方程组

consistent:有解 opp. inconsistent

equivalent:two linear systems have same solution

coefficient:matrix of left side of the equal

augmented:both side

\(m\times n\) （ \(m\) by \(n\) ）(rows come first)

replacement：加上其他行的倍数

interchange 交换

scaling 乘

3 elementary row operations

transform to the same matrix by these 3 opt,called row equivalent

\[\left[\begin{array}{llll} a_{1,1} &a_{2,1} &a_{3,1} &b_{1}\\ a_{1,2} &a_{2,2} &a_{3,2} &b_{2}\\ a_{1,3} &a_{2,3} &a_{3,3} &b_{3} \end{array}\right] \text{is a 3 by 4 augumented matrix} \]

（为啥不说是guassian elimination

a rectangular is in echelon form if it has the following tree examples:

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. all entries in a column below a leading entry are zeros

nonzero row: a row that contains at least one nonzero entry

leading entry of a row the leftmost nonzero entry

echelon form（倒梯形

echelon pattern（as following）:

Row Reduction Algorithm to form reduce echelon form

also have：

4. the leading entry in each nonzero row is one
5. the leading entry is the only nonzero entry in its column

e.g.:

\[\left[\begin{array}{llll} 1 &0 &-5 &1\\ 0 &1 &1 &4\\ 0 &0 &0 &0 \end{array}\right] \]

reduce echelon form are unique(not equivalent to any other reduce echelon) !!!!(1开头的倒三角矩阵)

the pivot column

pivot entry

the top of the leftmost position nonzero column is first pivot position

pivot:拿来消元的元素

pivot column:要消元的行

pivot: a pivot is an nonzero number in a pivot position that is used to create zeros via row operations

Section 1.2: Row Reduction and Echelon Forms

回代消元

basic variables are entries that are corresponding to the pivot column

free variables : otherwise

parametric

Theorem: Existence and Uniqueness Theorem

case1

\[\text{existtence} \Leftrightarrow \text{no row of the form } [0,\cdots,0,b] \\ \text{because its equation form is }0 = b \]

case2

\[\text{unique solution}\Leftrightarrow \text{no free variables} \]

case3

\[\text{infinitely many solutions} \Leftrightarrow \text{have free variables} \]

Finally we give a general solution,it gives all the representation of the basics variables just as following:

\[\left\{ \begin{align*} x_1 &= ... \\ x_3 &= ...\\ x_5 &= ... \end{align*} \right. \]

Section 1.3: Vector Equations

definition:vector

• A matrix with only one column is called a column vector, or simply a vector.

vectors in \(\mathbb{R}^n\) just like \(w = \left.\begin{bmatrix}w_1\\w_2\\...\end{bmatrix}\right\}\text{n rows}\)

scalars : 标量

definition:linear combination

\[y = c_1x_1+\cdots+c_nv_n \]

Remarks:marks

1. we always use \(\sim\) to represent row equivalent.