What exactly is a [linear algebra] vector?

What exactly is a vector?

Physics Professional Perspective

From a physics professional point of view, a vector is an arrow in space, and the characteristics that determine the vector are its length and its direction. As long as these two characteristics are the same, you can freely move a vector and keep it unchanged.

A vector in a plane is two-dimensional, while a vector in the space we live in is three-dimensional.

Computer professional perspective

From a computer professional perspective, a vector is an ordered list of numbers.

For example, assuming an analysis of house prices, a house can be modeled with a two-dimensional vector. The first number indicates the size of the house and the second number indicates the price.

Here, "vector" is just a fancy word for "list".

The reason this vector is two-dimensional is because the list is of length 2.

Mathematics Professional Perspective

From a mathematical perspective, a vector can make anything, as long as it makes sense.

Coordinates of a 2D vector

The coordinates of a vector consist of a logarithm that guides you from the origin (the start of the vector) to its tip (the end of the vector).

The first number tells you along $How\; far\; along\; the\; x$ -axis, the second number tells you along$How\; far\; the\; y$ -axis goes.

For example, the following vector is along $x$ - axis go$-2.0$ units of length,$then$ along$yy$$y$ axis goes$-1.5unit$ length$.$When the length is negative, it is walking in the opposite direction to the positive direction.

Coordinates of a 3D vector

On the basis of the two-dimensional coordinate system, add perpendicular to $x$ - axis and$The\; third\; axis\; of\; the\; y$ -axis, call it$z-$ axis.

In this case, each vector has an ordered tri-primitive array corresponding to it. The first number tells you along $How\; far\; along\; the\; x$ -axis, the second number tells you along$how\; far\; the\; y$ axis goes, the third number tells you along the$How\; far\; the\; z$ -axis goes.

Vector Addition and Vector Multiplication

vector addition

Assuming the addition of two vectors in the figure, we evaluate the second vector so that its start point coincides with the end point of the first vector.

Then draw a vector that starts from the start of the first vector and points to the end of the second vector.

This vector is their sum.
Numerically, the coordinates of the first vector are $\left[\begin{array}{l}1\\ 2\end{array}\right]$ , the coordinates of the second vector are$\left[\begin{array}{l}3\\ -1\end{array}\right]$ .

It's not hard to find that the sum vector is equal to moving to the right first$(1+3)$ steps, then move up$(2-1)$ step.

So the sum vector is:
$$\left[\begin{array}{l}1\\ 2\end{array}\right]+\left[\begin{array}{l}3\\ -1\end{array}\right]=\left[\begin{array}{l}1+3\\ 2+(-1)\end{array}\right]$$

The addition of vectors is to add up the corresponding numbers.
$$\left[\begin{array}{l}{x}_1\\ y_1\end{array}\right]+\left[\begin{array}{l}{x}_2\\ y_2\end{array}\right]=\left[\begin{array}{l}{x}_1+x_2\\ y_1+y_2\end{array}\right]$$

vector multiplication

Vector multiplication is to stretch the vector n times its original size.
For example: nn


here$n$ does not have the characteristics of a vector, it is calleda scalar.

Coordinate System

In general, xxA vector of length 1 in the positive direction of the$x$$A\; vector\; with\; a\; length\; of\; 1\; in\; the\; positive\; direction\; of\; the\; y$ -axis forms a coordinate system, but we can also customize these two unit vectors to change our coordinate system.

Zhang Cheng's space

$\vec{v}$与$\vec{w}$The vector set composed of all linear combinations is called "the space of Zhangcheng".

当$\vec{v}$与$\vec{w}$When they are not collinear, the space of Zhang Cheng is a plane.

当$\vec{v}$与$\vec{w}$When collinear, the space of Zhang Cheng is a straight line.

当$\vec{v}$与$\vec{w}$Both are $\vec{0}$, the space of Zhang Cheng is the origin.

Of course, this also applies to 3D coordinate systems.

$\vec{v}$与$\vec{w}$与$\vec{u}$The vector set composed of all linear combinations is called "the space of Zhangcheng".