6.78 min read

Weight Finding in an Orthonormal Basis

If $\{\mathbf{u}_1, \ldots, \mathbf{u}_n\}$ is an orthonormal basis for a subspace $V$ , then any $\mathbf{v} \in V$ can be written uniquely as $\mathbf{v} = c_1 \mathbf{u}_1 + \cdots + c_n \mathbf{u}_n$ . Finding the coefficients $c_i$ (the coordinates of $\mathbf{v}$ in this basis) normally requires solving a system of equations.

With an orthonormal basis, however, there is no solving needed. Taking the dot product of both sides with $\mathbf{u}_i$ collapses all terms to zero except the $i$ -th one: $c_i = \mathbf{u}_i \cdot \mathbf{v}$ . In matrix form, the whole coordinate vector is $U^t \mathbf{v}$ .

This is the key computational advantage of orthonormal bases: finding coordinates is just a matrix-vector multiplication with $U^t$ , not a linear system solve. The formula $\mathbf{v} = U(U^t \mathbf{v})$ says: "multiply by $U^t$ to get coordinates, then multiply by $U$ to reconstruct $\mathbf{v}$ ."

Formal View

Theorem 6.13 — Weight Finding

Let

\{\mathbf{u}_1, \ldots, \mathbf{u}_n\}

be an orthonormal basis for a subspace

V \subseteq \mathbb{R}^m

, and let

U

be the

m \times n

matrix with these columns. For any

\mathbf{v} \in V

, the coordinate vector in this basis is

\mathbf{v}_U := U^t \mathbf{v},

and

\mathbf{v} = U \mathbf{v}_U = U U^t \mathbf{v}

Proof: let $\mathbf{v} = \sum_j c_j \mathbf{u}_j$ . Then $U^t \mathbf{v} = U^t \sum_j c_j \mathbf{u}_j = \sum_j c_j (U^t \mathbf{u}_j) = \sum_j c_j \mathbf{e}_j = \mathbf{v}_U$ .

Why This Matters

Coordinate extraction via the transpose is what makes orthonormal bases so computationally powerful.

Fast Fourier Transform computes dot products with complex exponentials to find Fourier coefficients in $O(n \log n)$
MRI imaging uses orthonormal wavelets or sines as basis functions and extracts coefficients without solving any systems
Principal component analysis projects data onto orthonormal principal components via the same formula $U^t \mathbf{v}$

Learning Resources

Projections and orthonormal bases

MIT OpenCourseWare

Strang explains how orthonormal bases make projections and coordinate extraction trivial.

49 min

Change of basis

3Blue1Brown

Visual explanation of coordinates in a new basis and why orthonormal bases are special.

12 min

Quiz

Question 1

If $U$ has orthonormal columns and $\mathbf{v} \in \text{Col}(U)$ , what is $UU^t \mathbf{v}$ ?

Question 2

For a vector $\mathbf{v}$ in the column space of $U$ (orthonormal columns), the $i$ -th coordinate in the $U$ -basis is $\mathbf{u}_i \cdot \mathbf{v}$ .

Common Mistakes

Trying to find coordinates by solving $U\mathbf{c} = \mathbf{v}$ as a linear system when an orthonormal basis is available — just compute $U^t \mathbf{v}$ .
Applying the weight-finding formula $U^t \mathbf{v}$ to a vector $\mathbf{v}$ not in $\text{Col}(U)$ and expecting $UU^t\mathbf{v} = \mathbf{v}$ — this only works if $\mathbf{v} \in \text{Col}(U)$ .