Vectors are important parts of linear algebra and are used in many areas like math, engineering, physics, and computer science. To get a good grip on linear algebra and what it can do, it's important to understand what vectors are and their main features.

1. What is a Vector?

A vector is a math object that has two main things: size (or length) and direction. In linear algebra, we can represent vectors in something called n-dimensional space, which is written as $\mathbb{R}^n$.

For instance, a vector in 3D space ($\mathbb{R}^3$) looks like this:

$$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$$

Here, $v_1$, $v_2$, and $v_3$ are the parts of that vector.

2. Types of Vectors

There are different kinds of vectors:

• Column Vectors: These are shown as a single column. For example: $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$

• Row Vectors: These are shown as a single row. For example: $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

• Zero Vector: This vector has all its parts equal to zero. It’s represented as $\mathbf{0}$, like this: $\begin{bmatrix} 0 \\ 0 \\ ... \\ 0 \end{bmatrix}$

• Unit Vector: This is a special vector that has a length of 1. It is used to show direction.

3. Vector Operations

Here are some key things you can do with vectors:

• Addition: You can add two vectors by adding their parts together. For example, if $\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix},$ then:

  $$\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \end{bmatrix}$$

• Scalar Multiplication: If you have a number (called a scalar), you can multiply it by a vector. If $c$ is the number and $\mathbf{v}$ is the vector, it looks like this:

  $$c \mathbf{v} = \begin{bmatrix} c v_1 \\ c v_2 \end{bmatrix}$$

• Dot Product: This combines two vectors and gives you a number (called a scalar). For vectors $\mathbf{u}$ and $\mathbf{v}$, the dot product is:

  $$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2$$

This helps to find angles between the vectors.

4. Properties of Vectors

Vectors have some important properties:

• Commutative Property: This means you can add vectors in any order:

  $$\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$$

• Associative Property: This means when you add three vectors, the way you group them doesn’t matter:

  $$(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$$

• Distributive Property: If you multiply a vector addition by a number, it works like this:

  $$c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$$

5. Magnitude and Direction

The magnitude, or length, of a vector $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ can be found using this formula:

$$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}$$

To find the direction of a vector, we can normalize it, which means dividing it by its length to get a unit vector.

6. Vector Spaces

Vectors are the main parts of vector spaces. These are important structures in linear algebra. A vector space must meet four requirements:

1. It should be closed under addition.
2. It should be closed under scalar multiplication.
3. There must be a zero vector.
4. Each vector should have an opposite vector.

Understanding these basics about vectors is very important for learning more complicated topics in linear algebra. This includes things like matrix operations, transformations, eigenvalues, and uses in multivariable calculus and differential equations.