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Coordinate Vectors
Relative to a Basis

Learn how every vector gets a unique "address" once you choose a basis: its coordinate vector is the list of scalars that reconstruct it from the basis vectors.

Linear Algebra

What Is a Basis?

A basis for a vector space \(V\) is a set of vectors \(\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n\}\) that is both linearly independent and spans \(V\). Every vector in \(V\) can be written as a unique linear combination of these basis vectors.

Linearly Independent
No basis vector can be written as a combination of the others.
Spanning
Every vector in the space can be reached using the basis vectors.

The Coordinate Vector Idea

Given a basis \(\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n\}\) and a vector \(\mathbf{x} \in V\), there exist unique scalars \(c_1, c_2, \dots, c_n\) such that:

The coordinate vector of \(\mathbf{x}\) relative to \(\mathcal{B}\) is the column vector of those scalars:

Visual Intuition in \(\mathbb{R}^2\)

\(\mathbf{b}_1\) \(\mathbf{b}_2\) \(\mathbf{x}\) 2 1
\([\mathbf{x}]_{\mathcal{B}} = \begin{bmatrix}2\\1\end{bmatrix}\)
relative to \(\mathcal{B}\)
O

The Unique Representation Theorem

Why unique? Suppose two representations exist. Then subtracting them gives a nontrivial linear combination of the basis vectors equaling \(\mathbf{0}\), contradicting linear independence.

Coordinate Map as Matrix Equation

If we form a matrix \(P_{\mathcal{B}}\) whose columns are the basis vectors:

Then the equation \(\mathbf{x} = c_1\,\mathbf{b}_1 + \cdots + c_n\,\mathbf{b}_n\) becomes:

The matrix \(P_{\mathcal{B}}\) is called the change-of-coordinates matrix from \(\mathcal{B}\) to the standard basis. Its inverse converts standard coordinates back into \(\mathcal{B}\)-coordinates.

Standard vs. Non-Standard Basis

Standard Basis \(\mathcal{E}\) Non-Standard Basis \(\mathcal{B}\)
Vectors in \(\mathbb{R}^2\) \(\mathbf{e}_1 = \begin{bmatrix}1\\0\end{bmatrix}, \mathbf{e}_2 = \begin{bmatrix}0\\1\end{bmatrix}\) Any two linearly independent vectors
Coordinate vector \([\mathbf{x}]_{\mathcal{E}} = \mathbf{x}\) (the vector itself) \([\mathbf{x}]_{\mathcal{B}} = P_{\mathcal{B}}^{-1}\,\mathbf{x}\)
Matrix \(P\) \(I\) (identity) \(P_{\mathcal{B}} = [\mathbf{b}_1 \mid \mathbf{b}_2]\)
Computation Trivial: read off entries Solve a linear system or invert \(P_{\mathcal{B}}\)

How to Find Coordinate Vectors

1
Identify the basis \(\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n\}\) and the target vector \(\mathbf{x}\).
2
Set up the equation \(\mathbf{x} = c_1\,\mathbf{b}_1 + c_2\,\mathbf{b}_2 + \cdots + c_n\,\mathbf{b}_n\), which is the system \(P_{\mathcal{B}}\,\mathbf{c} = \mathbf{x}\).
3
Solve for the weights \(c_1, c_2, \dots, c_n\) by row reducing \([P_{\mathcal{B}} \mid \mathbf{x}]\) or computing \(P_{\mathcal{B}}^{-1}\,\mathbf{x}\).
4
Assemble the coordinate vector: \([\mathbf{x}]_{\mathcal{B}} = \begin{bmatrix}c_1\\c_2\\\vdots\\c_n\end{bmatrix}\).

Example 1: Simple Basis in \(\mathbb{R}^2\)

Let \(\mathcal{B} = \left\{\begin{bmatrix}1\\2\end{bmatrix},\,\begin{bmatrix}3\\1\end{bmatrix}\right\}\) and \(\mathbf{x} = \begin{bmatrix}7\\8\end{bmatrix}\). Find \([\mathbf{x}]_{\mathcal{B}}\).

Example 2: Using the Inverse Matrix

Let \(\mathcal{B} = \left\{\begin{bmatrix}2\\1\end{bmatrix},\,\begin{bmatrix}1\\3\end{bmatrix}\right\}\) and \(\mathbf{x} = \begin{bmatrix}5\\10\end{bmatrix}\).

Example 3: Coordinates in \(\mathbb{R}^3\)

Let \(\mathcal{B} = \left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix}\right\}\), \(\mathbf{x} = \begin{bmatrix}2\\3\\1\end{bmatrix}\). Find \([\mathbf{x}]_{\mathcal{B}}\).

Why Coordinate Vectors Matter

Simplify Computations
A clever choice of basis can turn complex operations into simple ones: diagonalization, Jordan forms, etc.
Abstract Spaces
Polynomials, matrices, and function spaces have bases too. Coordinates let us use \(\mathbb{R}^n\) tools on any finite-dim space.
Linear Transformations
The matrix of a linear map depends on the choice of basis: coordinate vectors are the bridge.

Beyond \(\mathbb{R}^n\): Polynomials

The space \(\mathcal{P}_2\) (polynomials of degree \(\leq 2\)) has standard basis \(\mathcal{B} = \{1, t, t^2\}\). For \(p(t) = 3 + 5t - 2t^2\):

With a different basis, say \(\mathcal{C} = \{1, 1+t, 1+t+t^2\}\), the coordinate vector would be different. Every basis assigns a unique "address" to the same polynomial.

Linearity of the Coordinate Map

The map \(\mathbf{x} \mapsto [\mathbf{x}]_{\mathcal{B}}\) preserves addition and scalar multiplication:

1
Addition: \([\mathbf{x} + \mathbf{y}]_{\mathcal{B}} = [\mathbf{x}]_{\mathcal{B}} + [\mathbf{y}]_{\mathcal{B}}\)
2
Scalar multiplication: \([r\,\mathbf{x}]_{\mathcal{B}} = r\,[\mathbf{x}]_{\mathcal{B}}\)
3
One-to-one and onto: Different vectors get different coordinate vectors, and every vector in \(\mathbb{R}^n\) is a coordinate vector for some \(\mathbf{x}\).

Common Mistakes

✕ Wrong Order
The entries of \([\mathbf{x}]_{\mathcal{B}}\) must follow the order of vectors in \(\mathcal{B}\). Swapping \(\mathbf{b}_1\) and \(\mathbf{b}_2\) changes the coordinate vector.
✕ Confusing \(\mathbf{x}\) with \([\mathbf{x}]_{\mathcal{B}}\)
The vector \(\mathbf{x}\) (in standard coordinates) is not the same as its coordinate vector relative to \(\mathcal{B}\), unless \(\mathcal{B}\) is the standard basis.
✕ Inverting the Wrong Matrix
To go from \(\mathbf{x}\) to \([\mathbf{x}]_{\mathcal{B}}\), multiply by \(P_{\mathcal{B}}^{-1}\), not \(P_{\mathcal{B}}\). The columns of \(P_{\mathcal{B}}\) are the basis vectors, not the coordinate vectors.

Summary of Key Facts

Concept Formula / Meaning
Coordinate vector \([\mathbf{x}]_{\mathcal{B}} = \begin{bmatrix}c_1\\\vdots\\c_n\end{bmatrix}\) where \(\mathbf{x} = \sum c_i\,\mathbf{b}_i\)
Change-of-coord. matrix \(P_{\mathcal{B}} = [\mathbf{b}_1 \mid \cdots \mid \mathbf{b}_n]\)
From standard to \(\mathcal{B}\) \([\mathbf{x}]_{\mathcal{B}} = P_{\mathcal{B}}^{-1}\,\mathbf{x}\)
From \(\mathcal{B}\) to standard \(\mathbf{x} = P_{\mathcal{B}}\,[\mathbf{x}]_{\mathcal{B}}\)
Linearity \([\mathbf{x}+\mathbf{y}]_{\mathcal{B}} = [\mathbf{x}]_{\mathcal{B}}+[\mathbf{y}]_{\mathcal{B}}\), \([r\mathbf{x}]_{\mathcal{B}} = r[\mathbf{x}]_{\mathcal{B}}\)
Standard basis special case \([\mathbf{x}]_{\mathcal{E}} = \mathbf{x}\)

Question 1 of 5

Let \(\mathcal{B} = \left\{\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}0\\1\end{bmatrix}\right\}\) (the standard basis) and \(\mathbf{x} = \begin{bmatrix}3\\-2\end{bmatrix}\). What is \([\mathbf{x}]_{\mathcal{B}}\)?

Question 2 of 5

Let \(\mathcal{B} = \left\{\begin{bmatrix}1\\1\end{bmatrix},\begin{bmatrix}1\\-1\end{bmatrix}\right\}\) and \(\mathbf{x} = \begin{bmatrix}4\\2\end{bmatrix}\). What is \([\mathbf{x}]_{\mathcal{B}}\)?

Question 3 of 5

If \([\mathbf{x}]_{\mathcal{B}} = \begin{bmatrix}2\\-1\end{bmatrix}\) and \(\mathcal{B} = \left\{\begin{bmatrix}3\\0\end{bmatrix},\begin{bmatrix}1\\2\end{bmatrix}\right\}\), what is \(\mathbf{x}\) in standard coordinates?

Question 4 of 5

In the polynomial space \(\mathcal{P}_2\) with basis \(\mathcal{B} = \{1, t, t^2\}\), what is \([4 - t + 7t^2]_{\mathcal{B}}\)?

Question 5 of 5

Which of the following is true about the coordinate mapping \(\mathbf{x} \mapsto [\mathbf{x}]_{\mathcal{B}}\)?