Lecture 9 Basis

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Textbook:  Introduction to Linear Algebra, Fifth Edition (2016)

Textbook open resource: http://math.mit.edu/~gs/linearalgebra/

這是第一章的收官作，在第一部分中我詳細介紹了線性代數最初解決的問題——線性方程的解以及其結構。在進入第二章之前，我將回到線性代數的幾何意義上，更直觀地揭示代數學對實際問題簡潔而深刻的認識。

（本文不適合暗黑模式下食用哦！推薦橫屏閱讀。）

0. Review

The main task of part I is to solve linear equation vector space. You should determine whether

A vector space is determined by its basis, for example

Or in short, they are columns of identity matrix

1. Plane system

A plane equals to a two-dimensional space. Descartes coordinate (

Here 

For the general case, for example vector

Since the direction of mirror is perpendicular to unit vector

This transformation is called reflection matrix (Householder transformation):

Algebra is a powerful tool to understand geometry. In lecture 1, I proved two equations by considering two vectors  

where  

LHS(Left-hand side) of the first equation is the inner product between

Consider a general case. There are three points forming a  triangle. To calculate the area of this triangle, translate one of its vertex to the origin. Therefore:

And the area:

A much elegant way:

(Prove that two deteminants are equal.)

If you exchange Define the positive or negative value as the "direction of area" along z-axis:

Now I have extended the plane to three-dimensional space. symbolically:

Now consider two arbitrary points in space: These two points with origin will form a plane, which is also a subspace.

Defnition 2.1: The cross product of

The vector is perpendicular tp

The length of this vector satisfy:

Since

It is remarkable that the triple product equals to the volume of the box with sides

3. Orthogonal Bases and Gram-Schmidt

All bases discussed in section 1 are orthogonal, for example rotation matrix:

Moveover:

They are unit vectors.

Definition 3.1: The vectors

Here

Let the vectors

Choosing a proper basis can simplfy many calculations. For example, computing projection

If

For any projection

Exactly the generalized Pythagoras theorem but not necessary to choose the special basis

Gram-Schmidt Process

Now we need to find a way to create orthonormal vectors. Gram-Schmidt process is one possible way.

Suppose a basis

Here orthogonal to

You can check

Again, deplete

Therefore, the "efficient" part of

That is:

NOTICE: You should use

In principle, you can repeatedly do this process until

Once you get all

unify the basis.

Such a construction is to:

Distribute

It finally gives:

In matrix form:

Gram-Schmit: From independent basis

4. Exercise

Exercise 1.1: For

(Hint:  

Exercise 2.1: Prove

Exercise 3.2: In least square approximation, if