"Linear Algebra and its Applications"-Vector equation

This piece of paper is used to record vectors, vector equations and their relationship to linear equations.

Vector: Talking about vectors, what is the first impression in your mind? Is the high school physics "namely has the direction and the size" and "the scalar" the relative abstract concept? Or a tool that doesn't matter where planar parsing and spatial geometry are determined? Or is that the bad guy in "The God who stole the milk Dad 2"? In fact, these are one-sided vectors, we will nx1 matrix as a vector, with r^n to represent, that is, the vector r^n is actually a column number, each number represents the vector a component size, we can also write it as <a1,a2,a3,... an> form.

The geometric meaning of the special vector: now based on the generalization of the vector definition, we look back at the vector we have used, for two components of the vector r^2, in fact, is to represent a plane in the direction of a line segment, for three components of the vector r^3, in fact, is a space within the direction of a line segment.

The addition and subtraction of vectors: The most basic thing about the operation of a vector is that it is based on its n components, that is, the vector a = <a1,a2>,b = <b1,b2> for two components, and a + b = <a1+b1,a2+b2>. Smart readers may have thought, this is in fact with our high school physics in the mechanics of the so-called "orthogonal decomposition" is the echo of each other, and in fact, we can get familiar with the so-called "parallelogram principle", "Triangle law."

The algebraic properties of a more comprehensive vector are given in the table below.

Vector equation: Before introducing the vector equation, we first introduce a noun-linear combination. As a simple answer, for variable x, which can be written as x = c1x1 + c2x2 (x1,x2 is a variable, C1,C2 is a constant), then we call x a linear combination of X1 and x2.

So it's the same thing to get the vector.

In the following, we use an example to elicit the vector equation based on the concept of this linear combination.

It is natural to see that the so-called vector equation is the equation that takes the vector as a parameter (and possibly an unknown amount).

We make further conversions.

Is the scene we are familiar with, this is the problem that we solved in the last article AH--solve the linear equation group.

That is to say here, we can understand that for linear vector equation X1c1 + x2c2...xncn=b (x1,x2,... xn is constant, c1,c2,c3,... B is a vector), it is actually with the augmented matrix is [C1,c2,... cn,b] of the linear Equation Group general solution, This allows us to directly transform the vector equation into an augmented matrix to solve the problem.

Meaning of the span symbol: based on a linear combination of concepts, we remember that span (V1,v2,v3,..., vn) represents a collection of all linear combinations of v1~vn of these n vectors.

Based on this concept of the span notation, we can further discover that span (v) and span (v,u) have practical geometric meanings.

Based on the understanding of the geometric meanings of span (v,u), let's look at an example of this.

To sum up, we can find that at the beginning of this book, the introduction of linear algebra is mainly from its "instrumental", and strive to let the reader understand the understanding of vectors, linear equations, matrices, which facilitate the operation of the concept of the beneficial tools and mutual connection, in order to establish a solid foundation for the proof of the following various theorems.

"Linear Algebra and its Applications"-Vector equation