Properties of Vectors

Properties of Vectors

We define a vector in $\mathbb{R}^n$ to be an ordered tuple of real numbers, $\vec{x} = (x_1, x_2, \cdots, x_n).$ This is the algebraic representation of vector $\vec{x}$.

The geometric representation of vector $\vec{x}$ is, in an $n$ dimensional space, an arrow from the origin to the point $(x_1, x_2, \cdots, x_n)$.

A vector has a length and a direction

Let $\vec{x} = (x_1, x_2, \cdots, x_n) \in \mathbb{R}^n$. The magnitude, or length, $\vec{x}$ is denoted as $|| \vec{x} ||$ and is defined as:

\[||\vec{x}|| = \sqrt{ {x_1}^2 + {x_2}^2 \cdots + {x_3}^2 }\]

This is essentially the pythagorean theorem in $n$ dimensions; in $\mathbb{R}^2$ the magnitude of a vector corresponds to the length of the hypotonuse of a right triangle whose other sides are of length $x_1$ and $x_2.$

The direction of a vector can be specified by the angle between it and some fixed reference, such as the $x$-axis.

The zero vector $(0, 0, \cdots, 0)$ is denoted as $\vec{0}$ and has no direction. Two vectors are equal if they have the same coordinates (or equivalently, the same length and direction).

We can multiple vectors by a scalar. Given the scalar $c$:

\[c \vec{x} = (c x_1, c x_2, \cdots, c x_n)\]

If a vector has a length of 1, i.e. if $||x|| = 1$, we say the vector is a unit vector.

We perform vector addition by adding two vectors $\vec{x} = (x_1, x_2, \cdots, x_n)$ and $\vec{y} = (y_1, y_2, \cdots, y_n)$ according to the following rule:

\[\vec{x} + \vec{y} = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n)\]

that is, by making a new vector where the coordinates are the sums of the respecive coordinates in the vectors being summed.

Similarly, vector subtraction can be performed as:

\[\vec{x} - \vec{y} = (x_1 - y_1, x_2 - y_2, \cdots, x_n - y_n)\]

Two vectors $\vec{x}$ and $\vec{y}$ are said to be parallel if $\vec{x}$ is a scalar multiple of $\vec{y}$, i.e., if there exists some scalar $c$ where $\vec{x} = c \vec{y}.$

Let $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n} \in \mathbb{R}^n$ and $c_1, c_2, \cdots, c_3 \in \mathbb{R}.$ Then, the vector

\[\vec{v} = c_1 \vec{v_1} + c_2 \vec{v_2} + \cdots + c_n \vec{v_n}\]

is called a linear combination of $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}.$

Let $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n} \in \mathbb{R}^n.$ The set of all linear combinations of $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}$ is called their span, denoted $\text{Span}{\left(\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}\right)}.$

That is:

\[\text{Span}{\left(\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}\right)} = \left\{ \vec{v} \in \mathbb{R}^n : \vec{v} = c_1 \vec{v_1} + c_2 \vec{v_2} + \cdots + c_n \vec{v_n} ~ \text{for some scalars} ~ c_1, c_2, \cdots, c_n \right\}\]