Introduction to Vectors and Linear Algebra

A vector is a quantity characterized by having both a magnitude and a direction.
A scalar quantity is characterized solely by magnitude, lacking any specified direction.
Information such as "moving at $5\,mph$ " is not a vector because it only specifies a magnitude. This is referred to as speed, which is a scalar quantity.
If a specific direction is added, such as "moving at $5\,mph$ due East," the quantity becomes a vector. In the context of motion, this combination of magnitude and direction is known as velocity.

In a two-dimensional space, a vector can be represented visually as an arrow.
The length of the arrow corresponds to the magnitude of the vector. For example, a vector representing a velocity of $5\,mph$ can be drawn as an arrow that is $5$ units long.
The direction in which the arrow points signifies the vector's direction.
When using a traditional two-dimensional coordinate system:
- The positive horizontal direction is typically designated as East.
- The negative horizontal direction is West.
- The positive vertical direction is North.
- The negative vertical direction is South.
An arrow starting at a point and extending $5$ units to the right along the horizontal axis represents a vector of magnitude $5$ pointing due East.

Vectors are defined exclusively by their magnitude and direction.
The specific starting point (initial position) of a vector does not change the identity of the vector itself when visualized.
Two arrows are considered equivalent vectors if they have the same length and point in the same direction, regardless of where they are placed on a coordinate plane.

When representing a vector as a variable, a lowercase letter is typically used.
In professional publishing (such as in textbooks), vector variables are often bolded.
In handwritten notes or notebooks, it is standard practice to place a small arrow on top of the lowercase letter to denote it as a vector.
While a vector can be described in words (e.g., " $5\,mph$ East"), this format is difficult to use for mathematical operations.

The standard mathematical way to represent a vector in two dimensions is by using two numbers that indicate the displacement in each dimension.
Vectors are often written as an ordered pair $(x, y)$ , where the first number represents horizontal movement and the second number represents vertical movement.
In the context of linear algebra, it is more typical to write vectors as column vectors: $\begin{pmatrix} x \\ y \end{pmatrix}$
For a vector moving $5$ units to the right and not moving vertically, the representations are:
- Ordered pair: $(5, 0)$
- Column vector: $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$
For a vector moving $3$ units in the horizontal direction and positive $4$ units in the vertical direction (designated as vector $\vec{a}$ ):
- Represents a shift of $3$ units right and $4$ units up.
- Column vector notation: $\vec{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$

The physical length or magnitude of a vector can be calculated using the Pythagorean theorem based on its horizontal and vertical components.
For a vector with components $(3, 4)$ , the magnitude is derived from the properties of a $3-4-5$ triangle.
Using the formula for the hypotenuse: $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
Therefore, a vector specified as $(3, 4)$ has an actual magnitude of $5$ .

Linear algebra allows for the mathematical extension of vector concepts beyond two dimensions.
While humans can easily visualize vectors in two or three dimensions, the human brain struggles to visualize vectors in four, five, or six dimensions.
Mathematical notation (such as coordinate lists or column vectors) is essential for dealing with vectors in higher dimensions, such as $n = 4$ , $n = 5$ , or even $n = 20$ , where drawing an arrow is impossible.
The rules of linear algebra that apply in two dimensions remain consistent as they extend into these more abstract, high-dimensional spaces.