Question

Let \(\left\\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{n}\right\\}\) be an orthogonal basis of \(V\). If \(\mathbf{v}\) and \(\mathbf{w}\) are in \(V\), show that $$ \langle\mathbf{v}, \mathbf{w}\rangle=\frac{\left\langle\mathbf{v}, \mathbf{f}_{1}\right\rangle\left\langle\mathbf{w}, \mathbf{f}_{1}\right\rangle}{\left\|\mathbf{f}_{1}\right\|^{2}}+\cdots+\frac{\left\langle\mathbf{v}, \mathbf{f}_{n}\right\rangle\left\langle\mathbf{w}, \mathbf{f}_{n}\right\rangle}{\left\|\mathbf{f}_{n}\right\|^{2}} $$

Short answer

The inner product is expressed as given through the coefficients in terms of the orthogonal basis.

Step-by-step solution

Understanding Inner Products with Orthogonal Basis

Given an orthogonal basis \(\{\mathbf{f}_1, \ldots, \mathbf{f}_n\}\) of the vector space \(V\), any vector \(\mathbf{v}\) in \(V\) can be expressed as \(\mathbf{v} = \sum_{i=1}^{n} c_i \mathbf{f}_i\), where the coefficients \(c_i\) are found using the projections \(c_i = \frac{\left\langle \mathbf{v}, \mathbf{f}_i \right\rangle}{\left\| \mathbf{f}_i \right\|^2}\).

Expressing Inner Product in Terms of Basis

For vectors \(\mathbf{v} = \sum_{i=1}^{n} c_i \mathbf{f}_i\) and \(\mathbf{w} = \sum_{j=1}^{n} d_j \mathbf{f}_j\), their inner product is given by: \(\left\langle \mathbf{v}, \mathbf{w} \right\rangle = \left\langle \sum_{i=1}^{n} c_i \mathbf{f}_i, \sum_{j=1}^{n} d_j \mathbf{f}_j \right\rangle \).

Applying Orthogonality

The orthogonality condition implies that \(\left\langle \mathbf{f}_i, \mathbf{f}_j \right\rangle = 0\) for \(i eq j\) and \(\left\langle \mathbf{f}_i, \mathbf{f}_i \right\rangle = \| \mathbf{f}_i \|^2\). Therefore, the inner product reduces to \(\left\langle \mathbf{v}, \mathbf{w} \right\rangle = \sum_{i=1}^{n} c_i d_i \| \mathbf{f}_i \|^2\).

Substituting Coefficients

Substitute \(c_i = \frac{\left\langle \mathbf{v}, \mathbf{f}_i \right\rangle}{\| \mathbf{f}_i \|^2}\) and \(d_i = \frac{\left\langle \mathbf{w}, \mathbf{f}_i \right\rangle}{\| \mathbf{f}_i \|^2}\) into the expression: \[ \left\langle \mathbf{v}, \mathbf{w} \right\rangle = \sum_{i=1}^{n} \frac{\left\langle \mathbf{v}, \mathbf{f}_i \right\rangle \left\langle \mathbf{w}, \mathbf{f}_i \right\rangle}{\| \mathbf{f}_i \|^2}. \] This matches the desired expression.

Key concepts

Orthogonal Basis

An orthogonal basis of a vector space is a special set of vectors. These vectors have a unique property: they are all orthogonal to each other. This means that the inner product of any two different vectors from this set is zero. Such a basis simplifies many calculations in linear algebra because it allows each component of a vector to be treated independently with respect to the basis.

• Every vector in the space can be uniquely expressed as a linear combination of the basis vectors.
• For computational purposes, using orthogonal bases helps in separating problems into simpler parts.

For example, if we have vectors \{\mathbf{f}_{1}, \mathbf{f}_{2}\} forming an orthogonal basis, then for a vector \( \mathbf{v} \), the coefficients in its linear combination representation are found using the inner product. This property makes orthogonal bases very useful in practice, especially in simplifying the calculation of inner products and in applications like computer graphics and machine learning.

Vector Projections

Vector projection involves taking one vector and projecting it onto another. This concept is particularly useful when dealing with bases. When a vector is projected onto an orthogonal basis, it breaks into components that align with each basis vector.
The projection of a vector \( \mathbf{v} \) onto a basis vector \( \mathbf{f}_i \) is calculated as follows:
\[\text{Projection of } \mathbf{v} \text{ onto } \mathbf{f}_i = \frac{\langle \mathbf{v}, \mathbf{f}_i \rangle}{\| \mathbf{f}_i \|^2} \mathbf{f}_i\]

• This formula involves the inner product of the vector with the basis vector.
• The division by the square of the norm of the basis vector ensures correct scaling.

This projection is crucial as it determines the coefficients \( c_i \) in the linear combination expression for the vector when using an orthogonal basis. This process helps in analyzing the individual contributions of each basis vector to the overall vector.

Orthogonality

Orthogonality is a core concept in vector spaces. Two vectors are orthogonal if their inner product is zero. This implies that the vectors are perpendicular to each other in the geometric sense. Orthogonality plays a fundamental role in simplifying problems in linear algebra.

• Orthogonality is the defining feature of an orthogonal basis.
• When vectors are orthogonal, their interactions are independent of one another, simplifying many mathematical operations.

When expressing the inner product in terms of an orthogonal basis, the orthogonality conditions \( \langle \mathbf{f}_i, \mathbf{f}_j \rangle = 0 \) for \( i eq j \) and \( \langle \mathbf{f}_i, \mathbf{f}_i \rangle = \| \mathbf{f}_i \|^2 \) are used. These conditions help reduce complex expressions to simpler sums of terms, as seen when determining the formula for the inner product of vectors \( \mathbf{v} \) and \( \mathbf{w} \) in an orthogonal basis:
\[\langle \mathbf{v}, \mathbf{w} \rangle = \sum_{i=1}^{n} \frac{\langle \mathbf{v}, \mathbf{f}_i \rangle \langle \mathbf{w}, \mathbf{f}_i \rangle}{\| \mathbf{f}_i \|^2}\]This mathematical property makes calculations easier and more efficient, especially in larger vector spaces.