# 1.1 Vectors and Linear Combination

## Abstract

This post series is an introduction to linear algebra. It’s simple and even naive, However, the important position of linear algebra won’t be changed. In artificial intelligence and other subjecs of engineering, linear algebra is the one of the most important foundation. So we need to review it again and agian, and it won’t waist our time at all, but, on the contrary, we can save a lot of time in our futher research. And Linear algebra theories can help us come up with some new ideas in our field.

**keywords**: *Vectors, Addition, Scalar Multiplication, Linear Combination, Picture of Linear Combination*

## The Map of Concepts

## Vectors

The first thing is about vectors. What is a vector? How do we develop a concept of vectors? What do vectors used for? All these questions are solved in linear algebra, and now let’s start with the most important concepts — vectors and their linear combinations — of the linear algebra.

### What a Vector Is

Well, when we are children, we have known 1,2,3…. etc., which they are so called numbers, and it’s used for presenting quantity – the amount of something there is. The numbers are a primary tool for human to describe the world in our daily life. But when our civilization went on to an advanced state, we found numbers – either real numbers or complex numbers – are limited. The numbers do fail to describe the things or actions that contain not only quantity but also other properties, such as directions.

To deal with these problems, at that time, most of them are in geometry and physics, someone developed vector, which is composed of more than one numbers. The idea is simple: if one number is not enough we can use more.

However, we got the foundation of the linear algebra. It’s not the matrises (you may heard before) or other things.It is vector. Vectors are the foundation of linear algebra, and linear algebra is found with vectors.

Forgive me for hammering this. But before we talk about space, vectors are the protagonist in our linear algebra novel.

The first part of our posts talk about some easy but important concepts of linear algebra, and I assume you have know vectors in high school.

Vectors are some numbers. Two numbers can produces a *two-dimensional* vector $\vec{v}$ (well, I’m sure you know this is a symbol of vector)

$$

\text{Column vector: } \vec{v}=\begin{bmatrix}v_1\newline v_2\end{bmatrix} \begin{aligned}&v_1= \text{first component}\newline &v_2 =\text{second component} \end{aligned} \tag{1.1.1}

$$

We write $\vec{v}$ as a column, not as a row. And the main point so far is that the single letter $\vec{v}$ (with an arrow on his head) is the pair of numbers $v_1$ and $v_2$

## Addition and Scalar Multiplication

“You can’t add apple and oranges.” Now, let’s look at how to add vector to vector.

### Vector Addition

$$

\vec{v}=\begin{bmatrix}v_1\newline v_2\end{bmatrix}

\text{ and }

\vec{w}=\begin{bmatrix}w_1\newline w_2\end{bmatrix}

\text{ add to }

\vec{v}+\vec{w}=

\begin{bmatrix}v_1+w_1\newline v_2+w_2\end{bmatrix} \tag{1.1.2}

$$

Aha, apple add to apple, and we get a new apple. But we can never add a *two-dimensional* vector to a vector whose dimension is not *two*. Subtraction is just a transformed addition, so we can get the vector Subtraction:

$$

\vec{v}=\begin{bmatrix}v_1\newline v_2\end{bmatrix}

\text{ and }

\vec{w}=\begin{bmatrix}w_1\newline w_2\end{bmatrix}

\text{ add to }

\vec{v}-\vec{w}=

\begin{bmatrix}v_1-w_1\newline v_2-w_2\end{bmatrix} \tag{1.1.3}

$$

They are the same thing – subtract a number is just add a minus number.

### Scalar Multiplication

$$

2\vec{v}=\begin{bmatrix}2v_1\newline 2v_2\end{bmatrix}

\text{ and }

-\vec{w}=\begin{bmatrix}-w_1\newline -w_2\end{bmatrix} \tag{1.1.4}

$$

Scalar multiplication is that a vector is multiplyed by a number, which is called scalar. The result of scalar multiplication is a new vector that is composed of components of origin vector multiplyed by the scalar.

The $-\vec{w}$ can be consider as $-1\cdot \vec{w}$. And a new idea come to us, what will happen if we add $\vec{v}$ and $-\vec{v}$. With the addition definition (1.1.2), we get:

$$

\begin{aligned}

&\vec{v}=\begin{bmatrix}v_1\newline v_2\end{bmatrix}

\text{ and }

-\vec{v}=\begin{bmatrix}-v_1\newline -v_2\end{bmatrix}\text{ then: }\newline

&\vec{v}+(-\vec{v})=

\begin{bmatrix}v_1-v_1\newline v_2-v_2\end{bmatrix}=

\begin{bmatrix}0\newline 0\end{bmatrix} \end{aligned}\tag{1.1.5}

$$

We call it zero vector, whose components are all 0’s, and its notation is not just $0$ but $\vec{0}$. And zero vector is not a number, it is always a vector with a constant-dimension! For instance, the result of (1.1.5) is $\vec{0}=\begin{bmatrix}0\newline 0\end{bmatrix}$ , while if $\vec{v}$ is a vector with three(or four five six…) components, the result will be a zero vector with three(or four five six…) zeroes. $\vec{0}=\begin{bmatrix}0\newline 0\end{bmatrix}$ and $\vec{0}=\begin{bmatrix}0\newline 0 \newline 0\end{bmatrix}$ are totally different.

Back to the addition, $\vec{v}+\vec{w}$ equals to $\vec{w}+\vec{v}$, for example:

$$

\vec{v}+\vec{w}=\begin{bmatrix}1\newline 5\end{bmatrix}+\begin{bmatrix}3\newline 3\end{bmatrix}=\begin{bmatrix}4\newline 8\end{bmatrix}\tag{1.1.6}

$$

and

$$

\vec{w}+\vec{v}=\begin{bmatrix}3\newline 3\end{bmatrix}+\begin{bmatrix}1\newline 5\end{bmatrix}=\begin{bmatrix}4\newline 8\end{bmatrix}\tag{1.1.7}

$$

**Please remember: adding vectors (addition) and multiplying by scalars (scalar multiplication) build linear algebra**

## Linear Combination

Combination of addition and multiplication have a special name, called “linear combination”. When we do a linear combination of $\vec{v}$ and $\vec{w}$, we multiply $\vec{v}$ by $c$ and multiply $\vec{w}$ by $d$ ;then add them $c\vec{v}+d\vec{w}$ together.

DEFINITION The sum of $c\vec{v}$ and $d\vec{w}$ is a linear combination of $\vec{v}$ and $vec{w}$

There are our special linear combinations, and the are: *sum*, *difference*, *zero*, and a *scalar multiple* $c\vec{v}$ :

$$

\begin {aligned}

1\vec{v}+1\vec{w} = & \text{ sum of vectors }\newline

1\vec{v}-1\vec{w} = & \text{ difference of vectors }\newline

0\vec{v}+0\vec{w} = & \text{ zero vector}\newline

c\vec{v}+0\vec{w} = & \text{ vector } c\vec{v} \text{ in the direction of } \vec{v}\newline

\end{aligned}

$$

In further discussion, we will often see the concept, “space” or “space of vector” which contain infinite vectors under some rules, **if the “space” are build on linear combination, the zero vector must in the space**. This big view, taking all the combinations of $\vec{v}$ and $\vec{w}$, is linear algebra at work.

### A Visual Vector

It’s a more easy way to view a vector that is drawing it on paper, but unfortunately, we can only draw two-dimensional vectors on paper. We use an arrow to represent $\vec{v}=\begin{bmatrix}1\newline 2\end{bmatrix}$, the arraw from $(0,0)$ points to $(1,2)$ like this:

Although we can understand addition, scalar multiplication and any other operations through this visual way, I don’t like this way, that’s easy but too easy to improve our imagination of vector and space. If you want to know more about how to use an arrow or two arrows to understand linear combination, you can search it on Google and find lots of things about that.

## Vectors in Three Dimensions

Of course, I will not draw a three dimensions vector on paper to show you how caculate addition of two 3-dimensions vectors. But one things we should pay attention to, that is the vector always be wrote as a column way like those above. However sometimes we write them in a row form to save space(not the “space” of vectors, but the space of paper), for example, we may write $\vec{v}=(1,2,3)$ to represent $\vec{v}=\begin{bmatrix}1\newline 2\newline 3\end{bmatrix}$ , they are the same thing. But $\vec{v}=(1,2,3)$ and $\vec{w}=\begin{bmatrix}1&2&3\end{bmatrix}$ are totally different things, the second one is called “row vector”

A three-dimensional linear combination are like this:

$$

c\vec{u}+d\vec{v}+e\vec{w}\tag{1.1.8}

$$

Where $\vec{u},\vec{v},\vec{w}$ are three-dimensonal vectors.

## The Important Questions

For one vector $\vec{u}$, its linear combinations are the multiples $c\vec{u}$ . For two vectors, the combinations are $c\vec{u}+d\vec{v}$. For three, the combinations are $c\vec{u}+d\vec{v}+e\vec{w}$. And results of these three kinds of combinations are 3-dimensional vectors. And we get the following three question:

- What is the picture of all combinations $c\vec{u}$
- What is the picture of all combinations $c\vec{u}+d\vec{v}$
- What is the picture of all combinations $c\vec{u}+d\vec{v}+e\vec{w}$

The answers are

1. line

2. plane

3. space(3-dimensions)

## Conclusion

- A two-dimensional vector has two components
- $\vec{v}+\vec{w}=\begin{bmatrix}v_1+w_1\newline v_2+w_2\end{bmatrix}$ , $c\vec{v}=\begin{bmatrix}cv_1\newline cv_2\end{bmatrix}$
- A linear combination of three vectors $\vec{u},\vec{v},\vec{w}$ are $c\vec{u}+d\vec{v}+e\vec{w}$ (no matter how many dimensions they are)
- Combinations $c\vec{u}$ typically fill a line
- Combinations $c\vec{u}+d\vec{v}$ typically fill a plane
- Combinations $c\vec{u}+d\vec{v}+e\vec{w}$ typically fill a space $\mathbb{R}^3$

## Reference

1.Strang G, Strang G, Strang G, et al. Introduction to linear algebra[M]. Wellesley, MA: Wellesley-Cambridge Press, 1993.

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