Coordinate¶

Definition¶

In $\mathbb{R}^n$, if

then $v_i$ is called the $i$th coordinate of $\boldsymbol{v}$.

Interpretation¶

The $i$th coordinate of $\boldsymbol{v}=(v_1,\ldots,v_n)$ is the number in position $i$ of the tuple: it says how far to go along the $i$th axis. Coordinates are the individual entries of a vector in real coordinate space, and two vectors of $\mathbb{R}^n$ are equal if and only if all their coordinates agree.

The figure shows that coordinates depend on the chosen basis. In the standard basis $\mathcal{E}=(\boldsymbol{e}_1,\boldsymbol{e}_2)$$({\mathbf{e}}_1,{\mathbf{e}}_2)$, the dashed grey lines drop from $\boldsymbol{w}$ to $3.8$ on the $x_1$-axis and to $2$ on the $x_2$-axis, so $[\boldsymbol{w}]_{\mathcal{E}}=(3.8,2)$ and $\boldsymbol{w}=3.8\boldsymbol{e}_1+2\boldsymbol{e}_2$$2{\mathbf{e}}_2$. Along the dashed blue path, the same vector is $\boldsymbol{w}=2\boldsymbol{b}_1+\boldsymbol{b}_2$${\mathbf{b}}_2$, so in the basis $\mathcal{B}=(\boldsymbol{b}_1,\boldsymbol{b}_2)$ it has coordinates $[\boldsymbol{w}]_{\mathcal{B}}=(2,1)$$(2,1)$.

In a general vector space the vectors need not be tuples, and coordinates are defined through a basis: they are the coefficients of the unique linear combination of the basis vectors that gives the vector. The standard basis of $\mathbb{R}^n$ makes those coefficients exactly the entries $v_i$, since $\boldsymbol{v}=v_1\boldsymbol{e}_1+\cdots+v_n\boldsymbol{e}_n$$v_n{\mathbf{e}}_n$. For polynomials of degree at most $2$ with the basis $(1,x,x^2)$, the coordinates of $3-x+4x^2$ are $(3,-1,4)$. In the plane, the Cartesian coordinates of a point are the coordinates of the arrow from the origin to that point in the standard basis.

A coordinate is a number, not a vector: $v_1$ is a scalar, while $v_1\boldsymbol{e}_1$ is the vector that goes $v_1$ steps along the first axis.

Key Results¶

• A vector in $\mathbb{R}^n$ has exactly $n$ coordinates.

Examples¶

• For $\boldsymbol{v}=(3,-1,4)$$(3,-1,4)$, the second coordinate is $-1$.

Prerequisites¶

• Vector Space

Related Concepts¶

• Basis
• Vector
• Euclidean Space
• Cartesian Coordinate System
• Inner Product

Sources¶

• Math II: Linear Algebra · Week 2

Aliases: coordinate tuple Keywords: component, coordinates Symbols: real numbers

Last updated: 20 October 2026 03:02