(a) Let 𝐮1, …, 𝐮n be an orthonormal basis of a...

(a) Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ be an orthonormal basis of a finite-dimensional inner product space $V$. Let $\mathbf{v}=c_1 \mathbf{u}_1+\cdots+c_n \mathbf{u}_n$ and $\mathbf{w}=d_1 \mathbf{u}_1+\cdots+d_n \mathbf{u}_n$ be any two elements of $V$. Prove that $\langle\mathbf{v}, \mathbf{w}\rangle=c_1 d_1+\cdots+c_n d_n$. (b) Write down the corresponding inner product formula for an orthogonal basis.

(a) Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ be an orthonormal basis of a finite-dimensional inner product space $V$. Let $\mathbf{v}=c_1 \mathbf{u}_1+\cdots+c_n \mathbf{u}_n$ and $\mathbf{w}=d_1 \mathbf{u}_1+\cdots+d_n \mathbf{u}_n$ be any two elements of $V$. Prove that $\langle\mathbf{v}, \mathbf{w}\rangle=c_1 d_1+\cdots+c_n d_n$.
(b) Write down the corresponding inner product formula for an orthogonal basis.

Key Concepts

Finite-Dimensional Inner Product Space

This is a vector space equipped with an inner product, which allows one to measure angles and lengths. Being finite-dimensional means that the vector space has a finite basis, ensuring that any vector in the space can be expressed as a linear combination of these basis vectors. The structure of such spaces supports concepts like orthogonality, projection, and the computation of inner products in a systematic way.

Orthonormal Basis

An orthonormal basis is a set of basis vectors that are both orthogonal and of unit length with respect to the inner product. The orthogonality property ensures that the inner product of any two distinct basis vectors is zero, while the norm condition guarantees that each basis vector has length one. This makes computations, especially those involving inner products and projections, particularly simple and elegant.

Expansion in Terms of a Basis

In any finite-dimensional vector space, any vector can be expressed as a linear combination of basis vectors. The coefficients in this expansion are found using the inner product when the basis is orthonormal, making the process straightforward. This expansion allows one to convert geometric problems into algebraic ones by working in the coordinate space defined by the basis.

Inner Product Formula

In a space with an orthonormal basis, the inner product of any two vectors can be computed directly as the sum of the products of their corresponding coordinates. This is due to the linearity of the inner product and the fact that the basis vectors have unit norm and are mutually orthogonal, simplifying the inner product computation to a convenient dot product-like formula.

Orthogonal Basis

An orthogonal basis is a set of basis vectors where each pair of distinct vectors is orthogonal. While not all vectors in an orthogonal basis need to be normalized (i.e., have unit length), the orthogonality property still greatly simplifies the computation of inner products and coordinates. The inner product of vectors expanded in an orthogonal basis includes weight factors corresponding to the norms of the basis vectors, ensuring consistency with the inner product definition.

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