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Orthonormal
Bases

Master the foundation of orthonormal bases and learn how to compute coordinates with simple dot products instead of solving systems of equations.

Linear Algebra

What is a Basis?

A basis for a vector space \(V\) is a set of vectors \(\{v_1, v_2, \ldots, v_n\}\) that is both linearly independent and spans all of \(V\). Every vector in \(V\) can be written as a unique linear combination of the basis vectors.

The scalars \(c_1, \ldots, c_n\) are the coordinates of \(x\) relative to that basis. Finding them usually requires solving a linear system, but an orthonormal basis gives us a shortcut.

Orthogonal vs Orthonormal

Orthogonal Set
Every pair of distinct vectors is perpendicular: \(\langle v_i, v_j \rangle = 0\) for \(i \neq j\).
Orthonormal Set
Orthogonal and every vector has unit length: \(\|v_i\| = 1\) for all \(i\).

Visualizing in \(\mathbb{R}^2\)

\(e_1\)
\(e_2\)
 1 1 O
\(x\)
\(c_1\)
\(c_2\)

The Coordinate Formula

Each coordinate \(c_i = \langle x, e_i \rangle\) is obtained by a single dot product, not by row-reducing a matrix. This is the key advantage of orthonormal bases.

Parseval's Identity

This says the squared length of a vector equals the sum of the squares of its coordinates: a generalized Pythagorean theorem. It holds because the cross terms \(\langle e_i, e_j \rangle\) vanish for \(i \neq j\).

General vs Orthonormal Coordinates

General Basis Orthonormal Basis
Find coordinates Solve \(Ax = b\) \(c_i = \langle x, e_i \rangle\)
Computation cost \(O(n^3)\) via row reduction \(O(n^2)\) via dot products
Length formula Requires inverse of Gram matrix \(\|x\|^2 = \sum |c_i|^2\)
Matrix of basis Any invertible matrix Orthogonal matrix: \(Q^TQ = I\)

Orthogonal Matrices

When you stack orthonormal basis vectors as columns of a matrix \(Q\), you get an orthogonal matrix with a remarkable property:

Preserves Lengths
\(\|Qx\| = \|x\|\) for all \(x\).
Preserves Angles
\(\langle Qx, Qy \rangle = \langle x, y \rangle\).
Determinant
\(\det(Q) = \pm 1\).

The Gram-Schmidt Process

Any basis can be converted into an orthonormal basis using Gram-Schmidt orthogonalization:

1
Start: Set \(u_1 = v_1\) and normalize: \(e_1 = u_1 / \|u_1\|\).
2
Subtract projections: \(u_k = v_k - \sum_{j=1}^{k-1} \langle v_k, e_j \rangle\, e_j\).
3
Normalize: \(e_k = u_k / \|u_k\|\).
4
Repeat steps 2-3 for \(k = 2, 3, \ldots, n\).

How to Compute Coordinates

1
Verify orthonormality: Check \(\langle e_i, e_j \rangle = 0\) for \(i \neq j\) and \(\|e_i\| = 1\).
2
Compute each coordinate: \(c_i = \langle x, e_i \rangle\) (one dot product per basis vector).
3
Write the expansion: \(x = c_1 e_1 + c_2 e_2 + \cdots + c_n e_n\).
4
Verify (optional): Check \(\|x\|^2 = c_1^2 + c_2^2 + \cdots + c_n^2\) (Parseval).

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

Let \(e_1 = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\) and \(e_2 = \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Find the coordinates of \(x = (3, 1)\).

\(e_1\)
\(e_2\)
\(x = (3,1)\)

Example 1: Verification

We found \(c_1 = 2\sqrt{2}\) and \(c_2 = -\sqrt{2}\). Let us verify using Parseval's identity.

Example 2: Coordinates in \(\mathbb{R}^3\)

Let the orthonormal basis be:

Find the coordinates of \(x = (1, 0, 2)\).

1
\(c_1 = \langle x, e_1 \rangle = \frac{1}{\sqrt{3}}(1 + 0 + 2) = \frac{3}{\sqrt{3}} = \sqrt{3}\)
2
\(c_2 = \langle x, e_2 \rangle = \frac{1}{\sqrt{2}}(1 - 0 + 0) = \frac{1}{\sqrt{2}}\)
3
\(c_3 = \langle x, e_3 \rangle = \frac{1}{\sqrt{6}}(1 + 0 - 4) = \frac{-3}{\sqrt{6}}\)

Example 2: Parseval Check

Notice we verified our answer without expanding the entire linear combination. This makes Parseval's identity a powerful error-checking tool.

Common Mistakes

✕ Forgetting to Normalize
Using \(c_i = \langle x, v_i \rangle\) when \(v_i\) is orthogonal but not unit length. You need \(c_i = \langle x, v_i \rangle / \|v_i\|^2\) for non-normalized vectors.
✕ Applying to Non-Orthogonal Bases
The dot-product formula only works for orthonormal bases. For a general basis you must solve \(Ax = b\).
✕ Incorrect Gram-Schmidt Projection
The projection of \(v_k\) onto \(e_j\) is a multiple of \(e_j\). Remember to subtract \(\langle v_k, e_j \rangle e_j\), not a term involving scaling \(v_k\).

Summary of Key Facts

Concept Formula / Property
Orthonormality condition \(\langle e_i, e_j \rangle = \delta_{ij}\)
Coordinate computation \(c_i = \langle x, e_i \rangle\)
Expansion \(x = \sum_{i=1}^n \langle x, e_i \rangle\, e_i\)
Parseval's identity \(\|x\|^2 = \sum |c_i|^2\)
Orthogonal matrix \(Q^T Q = I\), \(Q^{-1} = Q^T\)
Gram-Schmidt \(u_k = v_k - \sum \langle v_k, e_j \rangle e_j\)

Question 1 of 5

If \(\{e_1, e_2\}\) is an orthonormal basis and \(x = 3e_1 + 4e_2\), what is \(\|x\|\)?

Question 2 of 5

Let \(e_1 = \left(\frac{3}{5}, \frac{4}{5}\right)\) and \(e_2 = \left(-\frac{4}{5}, \frac{3}{5}\right)\). What is the coordinate \(c_1 = \langle x, e_1 \rangle\) for \(x = (6, 8)\)?

Question 3 of 5

Which property does not hold for an orthogonal matrix \(Q\)?

Question 4 of 5

In the Gram-Schmidt process, what is the purpose of the subtraction step \(u_k = v_k - \sum \langle v_k, e_j \rangle e_j\)?

Question 5 of 5

An orthonormal basis \(\{e_1, e_2, e_3\}\) yields coordinates \((2, -1, 3)\) for a vector \(x\). What is \(\|x\|^2\)?