Gilbert Strang's course notes on linear algebra - Lesson 1

The goal is to write notes so that everyone can understand : )

First we need to understand , The basic goal of linear algebra is ： Solve the problem of linear system

The first lesson mainly explains linear systems from two perspectives （ Transverse and longitudinal ： Row image Row Picture And column image Column Picture）

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First, we express the equations in matrix form ：

Suppose there are now Equations

Write it in matrix form as

【 Why can it be expressed like this ？

If you don't understand the above formula, review the dot multiplication of the matrix ：

The operation rule of matrix point multiplication is ：

set up A by m x n Matrix ,B by n x p Matrix , be A And B The product of is a matrix C = AB

1. C The dimensions are m x p
2. C Of the i That's ok j Column elements Cij from A Of the i Row elements and B Of the j The column elements are multiplied by each other and then added

Substitute the above example ：

A by 2 x 2 Matrix ,B by 2 x 1 Matrix , obtain C by 2 x 1 Matrix

C Medium "0" stay 1 That's ok 1 Column , from A The first line of elements [2, -1] And B The first column of elements [x, y] Elements are multiplied by each other ：

2x – y = 0, That is the first equation , The second equation is the same .

Other operations of matrix ：6.5 Matrix operation and its operation rules 】

Usually , We abbreviate the above example as Ax = b

A For the parameter matrix ,x Is the variable matrix ,b For the target

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Let's explain this equation set from the horizontal and vertical angles

1. Row image （Row Picture）

Row image means to understand the equations in rows , That is our traditional way of understanding

namely ： Find the intersection of two sets of relationships

We think 2x - y = 0 Depicted in x-y All points on the coordinate axis that conform to the equation ,-x + 2y = 3 Empathy

The solution is the intersection of two lines ：x = 1, y = 2

2. Column image （Column Picture）

Column image means to understand equations in columns , We have hardly understood equations in this way before

Let's look at the equation vertically and rewrite it as ：

x + y =

We think x individual add y individual You can get

namely ： Find a linear combination of a set of columns to express the target column

When you visualize it on the axis , It can be understood as x individual Vector plus y individual Vector can be obtained vector

（ Note that the coordinate axis at this time is not the original x-y Axis , Because of the original x、y The meaning of becomes " combination "）

u The vector is ,v The vector is , The target vector w Is the vector to the right of the equation

The knowable solution is ：1 individual u Vector and 2 individual v Vector combination can get the target vector w,x = 1,y = 2

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alike , In three-dimensional space, we can also use the angle of row image and column image to explain ：

Suppose there are now Equations

Write it as Ax = b In the form of ：

Parameter matrix A = ,b =

1. Understand by line image , Three planes can be drawn on the three-dimensional coordinate axis ：

The intersection of three planes is the solution

2. Understand by column image , Four vectors can be drawn on the three-dimensional coordinate axis

 

OA by ,OB by ,OC by ,OD by （OE And OC It cannot be displayed if it overlaps ）

In this case, because the target vector OD And one of the vectors OC coincidence , Are all , And the other two vectors OA、OB And vector OC There is no linear combination relationship , So the final solution is 0 individual OA vector +0 individual OB vector +1 individual OC vector =OD vector

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attach ： The tools used for drawing are ：Calculator Suite - GeoGebra

Gilbert Strang The teaching video of is （ Need scientific Internet ）：

https://www.youtube.com/watch?v=ZK3O402wf1c&list=PL49CF3715CB9EF31D&index=1