Vectors and Vector Operations

Definition Let $n$ be a positive integer. A vector in $\mathbb{R}^n$ is an ordered list of $n$ real numbers, called its components or entries. We write a vector as a column vector,

and $\mathbb{R}^n$ is the set of all such columns. Two vectors are equal when they have the same number of entries and agree in every position. Vectors get bold lowercase letters. In running text a tuple such as $(2, 1)$ stands for the column vector with entries $2$ and $1$.

Definition Fix an origin and coordinate axes. A point is a location. We name points with capital letters and give them by their coordinates, as in $P = (p_1, \dots, p_n)$. Points are written with parentheses too, so a capital letter or the word point marks such a tuple as a location rather than a column vector. For points $P$ and $Q$, the displacement from $P$ to $Q$ is the vector

endpoint minus starting point. Any arrow whose tip coordinates minus tail coordinates give this column represents the same vector. In particular, sliding an arrow to a new starting point without turning or stretching it does not change the vector it represents. Adding the displacement to the coordinates of $P$ gives the coordinates of $Q$.

Definition For vectors $\mathbf{u}$ and $\mathbf{v}$ in the same $\mathbb{R}^n$, the sum and the difference are formed entry by entry,

Vectors with different numbers of entries cannot be added or subtracted. In $\mathbb{R}^2$ and $\mathbb{R}^3$ the sum follows the tip-to-tail rule. Draw $\mathbf{u}$, then draw a copy of $\mathbf{v}$ with its tail at the tip of $\mathbf{u}$. The arrow from the tail of $\mathbf{u}$ to the tip of that copy represents $\mathbf{u} + \mathbf{v}$. When $\mathbf{u}$ and $\mathbf{v}$ are drawn from the same point, $\mathbf{u} - \mathbf{v}$ is the arrow from the tip of $\mathbf{v}$ to the tip of $\mathbf{u}$, since $\mathbf{v} + (\mathbf{u} - \mathbf{v}) = \mathbf{u}$. Addition is commutative and associative,

because addition of real numbers has these properties in each entry. Subtraction is not commutative. Each entry of $\mathbf{v} - \mathbf{u}$ is the negative of the matching entry of $\mathbf{u} - \mathbf{v}$, so the two differences agree only when $\mathbf{u} = \mathbf{v}$.

Definition A real number that multiplies a vector is called a scalar. For a scalar $c$ and a vector $\mathbf{v}$ in $\mathbb{R}^n$, the scalar multiple $c\mathbf{v}$ multiplies every entry by $c$,

The vector $(-1)\mathbf{v}$ is written $-\mathbf{v}$ and called the opposite of $\mathbf{v}$. In $\mathbb{R}^2$ and $\mathbb{R}^3$, for a nonzero vector $\mathbf{v}$ and a nonzero scalar $c$, the arrow of $c\mathbf{v}$ points the same way as $\mathbf{v}$ when $c > 0$ and the opposite way when $c < 0$, and it is $\lvert c \rvert$ times as long. For all vectors $\mathbf{u}, \mathbf{v}$ in $\mathbb{R}^n$ and scalars $c, d$, the distributive laws

hold, together with $c(d\mathbf{v}) = (cd)\mathbf{v}$ and $1\mathbf{v} = \mathbf{v}$. Each follows from the same law for real numbers, applied in every entry.

Definition The zero vector $\mathbf{0}$ of $\mathbb{R}^n$ is the vector whose $n$ entries are all $0$. For every $\mathbf{v}$ in $\mathbb{R}^n$,

So $\mathbf{0}$ is the identity for vector addition, and the opposite $-\mathbf{v}$ is the vector that cancels $\mathbf{v}$. Subtracting a vector is the same as adding its opposite, $\mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v})$. As a displacement, $\mathbf{0}$ is no change of position, the displacement from any point to itself. Its arrow has its tail and tip at the same point, so it has no direction.

Definition A data vector is a vector in $\mathbb{R}^n$ whose entries are $n$ measurements or counts listed in a fixed order, with a stated meaning and unit for each position. Data vectors are added, subtracted and scaled by the same entry by entry rules. A sum or difference of two data vectors answers a sensible question only when both use the same order, meanings and units, so that matching positions describe the same quantity.