Vector Operations in ℝⁿ

Intuition. A vector is two things at once: a list of numbers and an arrow. In $\mathbb{R}^2$ you can picture the vector $\mathbf{v} = (3, 2)$ as an arrow from the origin that goes $3$ units right and $2$ units up. The same idea works in $\mathbb{R}^3$ (an arrow in space) and in $\mathbb{R}^n$ for any $n$, where we can no longer draw it but the arithmetic is identical.

Formal definition. The space $\mathbb{R}^n$ is the set of all ordered lists of $n$ real numbers: $\mathbb{R}^n = \{\,(v_1, v_2, \ldots, v_n) : v_i \in \mathbb{R}\,\}.$ An element $\mathbf{v} = (v_1, v_2, \ldots, v_n)$ is called a vector, and each $v_i$ is its $i$-th component (also called a coordinate or entry). The number $n$ is the dimension. We write vectors in bold, $\mathbf{v}$, and use plain letters like $c$ for scalars (ordinary real numbers).

The zero vector. The special vector whose every component is $0$, $\mathbf{0} = (0, 0, \ldots, 0),$ is the zero vector. It is the arrow of length zero, and it plays the role of 'nothing to add.'

Equality. Two vectors are equal exactly when they live in the same $\mathbb{R}^n$ and agree in every component: $\mathbf{u} = \mathbf{v}$ means $u_i = v_i$ for all $i = 1, \ldots, n$.

Everything in this lesson is built from two moves, both performed one component at a time.

Vector addition. For $\mathbf{u} = (u_1, \ldots, u_n)$ and $\mathbf{v} = (v_1, \ldots, v_n)$ in the same $\mathbb{R}^n$, $\mathbf{u} + \mathbf{v} = (u_1 + v_1,\; u_2 + v_2,\; \ldots,\; u_n + v_n).$ You can only add vectors of the same dimension.

Scalar multiplication. For a scalar $c \in \mathbb{R}$ and a vector $\mathbf{v}$, $c\,\mathbf{v} = (c\,v_1,\; c\,v_2,\; \ldots,\; c\,v_n).$ Every component is multiplied by $c$.

Subtraction is just addition of a scaled vector: $\mathbf{u} - \mathbf{v} = \mathbf{u} + (-1)\mathbf{v} = (u_1 - v_1, \ldots, u_n - v_n)$.

Geometry.

• Addition (tip-to-tail). To form $\mathbf{u} + \mathbf{v}$, draw $\mathbf{u}$, then start $\mathbf{v}$ at the tip of $\mathbf{u}$; the sum is the arrow from the tail of $\mathbf{u}$ to the new tip. Equivalently, the parallelogram rule says $\mathbf{u}+\mathbf{v}$ is the diagonal of the parallelogram whose sides are $\mathbf{u}$ and $\mathbf{v}$.
• Scaling. Multiplying by $c > 1$ stretches a vector, $0 < c < 1$ shrinks it, and $c < 0$ flips it to point the opposite way while scaling its length by $|c|$. Multiplying by $0$ collapses any vector to $\mathbf{0}$.

A linear combination is what you get by scaling several vectors and adding the results. Given vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k$ in $\mathbb{R}^n$ and scalars $c_1, c_2, \ldots, c_k \in \mathbb{R}$, the vector $c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_k \mathbf{v}_k$ is a linear combination of $\mathbf{v}_1, \ldots, \mathbf{v}_k$ with coefficients (or weights) $c_1, \ldots, c_k$.

This single idea is the engine of linear algebra. For example, every vector in $\mathbb{R}^2$ is a linear combination of the standard basis vectors $\mathbf{e}_1 = (1,0)$ and $\mathbf{e}_2 = (0,1)$, because $(a, b) = a\,(1,0) + b\,(0,1) = a\,\mathbf{e}_1 + b\,\mathbf{e}_2.$ Asking 'is $\mathbf{b}$ a linear combination of these vectors, and with which coefficients?' turns into solving a system of linear equations — a theme you will meet again and again.

Because addition and scaling act one component at a time, they inherit the familiar arithmetic rules of real numbers. For all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and all scalars $c, d \in \mathbb{R}$:

• Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$.
• Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$.
• Additive identity: $\mathbf{u} + \mathbf{0} = \mathbf{u}$.
• Additive inverse: $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$, where $-\mathbf{u} = (-1)\mathbf{u}$.
• Distributivity over vector sums: $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$.
• Distributivity over scalar sums: $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$.
• Compatibility of scalars: $c(d\mathbf{u}) = (cd)\mathbf{u}$, and $1\,\mathbf{u} = \mathbf{u}$.

These are precisely the axioms that make $\mathbb{R}^n$ a vector space. Each one is proved by checking a single component and using the matching rule for real numbers.

Norm (magnitude). The norm of $\mathbf{v} = (v_1, \ldots, v_n)$ measures its length: $\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}.$ In $\mathbb{R}^2$ this is just the Pythagorean theorem: $\|(3,4)\| = \sqrt{9 + 16} = 5$. Scaling obeys $\|c\mathbf{v}\| = |c|\,\|\mathbf{v}\|$, matching the picture that multiplying by $c$ scales length by $|c|$.