Coordinate Vector

32. Coordinate Vector

Given a basis $𝐵 = {{\vec{𝑣}}_{1}, {\vec{𝑣}}_{2}, \dots, {\vec{𝑣}}_{𝑛}}$ of a vector space $𝑉$ , every $\vec{𝑥} \in 𝑉$ has a unique representation as a linear combination of the basis vectors:

\vec{𝑥} = 𝑐_{1} {\vec{𝑣}}_{1} + 𝑐_{2} {\vec{𝑣}}_{2} + \dots + 𝑐_{𝑛} {\vec{𝑣}}_{𝑛}

The coordinate vector of $\vec{𝑥}$ with respect to $𝐵$ collects those coefficients:

{[\vec{𝑥}]}_{𝐵} = [\begin{matrix} 𝑐_{1} \\ 𝑐_{2} \\ ⋮ \\ 𝑐_{𝑛} \end{matrix}]

The bracket-subscript notation ${[\vec{𝑥}]}_{𝐵}$ emphasizes that the coordinates depend on the basis.

32.1. The standard basis is implicit

In $ℝ^{𝑛}$ , when we write $\vec{𝑥} = [\begin{matrix} 3 \\ - 1 \\ 5 \end{matrix}]$ , we tacitly mean coordinates with respect to the standard basis $𝐸 = {{\vec{𝑒}}_{1}, \dots, {\vec{𝑒}}_{𝑛}}$ :

{[\vec{𝑥}]}_{𝐸} = [\begin{matrix} 3 \\ - 1 \\ 5 \end{matrix}]

When you change to a different basis, the same vector gets different coordinates.

Example

Let $𝐵 = {[\begin{matrix} 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \end{matrix}]}$ — a basis of $ℝ^{2}$ .

Express $\vec{𝑥} = [\begin{matrix} 3 \\ 1 \end{matrix}]$ in basis $𝐵$ . Solve:

𝑐_{1} [\begin{matrix} 1 \\ 1 \end{matrix}] + 𝑐_{2} [\begin{matrix} 1 \\ - 1 \end{matrix}] = [\begin{matrix} 3 \\ 1 \end{matrix}]

\begin{matrix} 𝑐_{1} + 𝑐_{2} = 3 \\ 𝑐_{1} - 𝑐_{2} = 1 \end{matrix}

→ $𝑐_{1} = 2$ , $𝑐_{2} = 1$ . So ${[\vec{𝑥}]}_{𝐵} = [\begin{matrix} 2 \\ 1 \end{matrix}]$ , while ${[\vec{𝑥}]}_{𝐸} = [\begin{matrix} 3 \\ 1 \end{matrix}]$ .

32.2. Why coordinates matter

Coordinates make abstract vectors computable: once we fix a basis, every vector is a column of numbers and every linear transformation is a matrix.

Different bases reveal different aspects:

Standard basis: most natural for $ℝ^{𝑛}$
Eigenbasis: makes the matrix of a linear transformation diagonal (when one exists — see Diagonalization)
Orthonormal basis: distances and angles match the standard formulas
Adapted basis: chosen to simplify a specific subspace structure

32.3. Connection to change of basis

When you switch between two bases $𝐵$ and $𝐵^{'}$ , the coordinate vectors are related by an invertible matrix — see Change of Basis.

32.4. See also

Basis
Change of Basis
Vector Space
Matrix Representation — coordinates of $𝑇 (\vec{𝑥})$ in terms of coordinates of $\vec{𝑥}$