Question

Let \(T: V \rightarrow W\) be any linear transformation and let \(B=\left\\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\\}\) and \(D=\left\\{\mathbf{d}_{1}, \ldots, \mathbf{d}_{m}\right\\}\) be bases of \(V\) and \(W\), respectively. If \(W\) is an inner product space and \(D\) is orthogonal, show that $$ M_{D B}(T)=\left[\frac{\left(\mathbf{d}_{i}, T\left(\mathbf{b}_{j}\right)\right\rangle}{\left\|\mathbf{d}_{i}\right\|^{2}}\right] $$ This is a generalization of Theorem \(10.3 .2 .\)

Short answer

The matrix of \( T \) relative to \( B \) and orthogonal \( D \) is \( \left[ \frac{\langle \mathbf{d}_i, T(\mathbf{b}_j) \rangle}{\| \mathbf{d}_i \|^2} \right] \).

Step-by-step solution

Understand the Problem

We are asked to show that the matrix representation of a linear transformation \( T \) with respect to the bases \( B \) and \( D \) is given by a specific expression when \( W \) is an inner product space and \( D \) is orthogonal. This requires us to utilize properties of linear transformations, inner products, and orthogonal sets.

Recall the Matrix Representation of a Linear Transformation

For a linear transformation \( T: V \to W \), the matrix \( M_{DB}(T) \) consists of elements \( a_{ij} \), which transform basis vectors \( \mathbf{b}_j \) from \( B \) into a linear combination of basis vectors \( \{ \mathbf{d}_i \} \) of \( D \). The element \( a_{ij} \) in an inner product space can be found using the relation \( T(\mathbf{b}_j) = a_{1j}\mathbf{d}_1 + a_{2j}\mathbf{d}_2 + \cdots + a_{mj}\mathbf{d}_m \).

Use the Properties of an Orthogonal Basis

Because \( D \) is an orthogonal basis of \( W \), each vector \( \mathbf{d}_i \) has an inner product \( \langle \mathbf{d}_i, \mathbf{d}_j \rangle = 0 \) for all \( i e j \), and \( \langle \mathbf{d}_i, \mathbf{d}_i \rangle = \| \mathbf{d}_i \|^2 \). This orthogonality simplifies finding the coefficients in the linear combination representing \( T(\mathbf{b}_j) \).

Calculate the Coefficients with Inner Products

We need to express \( T(\mathbf{b}_j) \) as a sum of the basis vectors in \( D \), \( T(\mathbf{b}_j) = a_{1j}\mathbf{d}_1 + a_{2j}\mathbf{d}_2 + \ldots + a_{mj}\mathbf{d}_m \), and to find the coefficients, we use the fact that \( a_{ij} = \frac{\langle \mathbf{d}_i, T(\mathbf{b}_j) \rangle}{\langle \mathbf{d}_i, \mathbf{d}_i \rangle} \). This is because \( \langle \mathbf{d}_i, T(\mathbf{b}_j) \rangle \) gives the projection of \( T(\mathbf{b}_j) \) onto \( \mathbf{d}_i \).

Conclude with the Desired Matrix Representation

By the previous step, since we are given \( W \) is an inner product space and \( D \) is orthogonal, each entry of the matrix is \( \frac{\langle \mathbf{d}_i, T(\mathbf{b}_j) \rangle}{\| \mathbf{d}_i \|^2} \). Thus, the matrix representation \( M_{DB}(T) \) is indeed \( \left[ \frac{\langle \mathbf{d}_i, T(\mathbf{b}_j) \rangle}{\| \mathbf{d}_i \|^2} \right] \), as required.

Key concepts

Understanding Inner Product Spaces

An inner product space is a crucial concept in linear algebra and functional analysis. It is essentially a vector space that has an additional structure enabling us to measure angles and lengths. This measurement is facilitated by the inner product, a function that maps two vectors to a scalar and satisfies several key properties: linearity, symmetry, and positive definiteness.

• **Linearity**: The inner product is linear in its first argument, which means \( \langle a\mathbf{x} + b\mathbf{y}, \mathbf{z} \rangle = a\langle \mathbf{x}, \mathbf{z} \rangle + b\langle \mathbf{y}, \mathbf{z} \rangle \).
• **Symmetry**: The function is symmetric, indicating that \( \langle \mathbf{x}, \mathbf{y} \rangle = \langle \mathbf{y}, \mathbf{x} \rangle \).
• **Positive Definiteness**: Finally, it is positive definite, meaning \( \langle \mathbf{x}, \mathbf{x} \rangle \geq 0 \), with equality if and only if \( \mathbf{x} = \mathbf{0} \).

These properties enable operations like finding angles between vectors, lengths of vectors (via norms), and orthogonal projections. Inner product spaces generalize these geometric notions beyond the familiar 2D and 3D spaces, making them indispensable in high-dimensional applications.

Exploring Orthogonal Bases

An orthogonal basis for a vector space is a set of vectors that are all orthogonal to each other, meaning each pair of distinct vectors in the set has an inner product of zero. Choosing an orthogonal basis offers many computational advantages:

• **Simplified Calculations**: Projections and decompositions of vectors are much easier and faster with orthogonal bases, as you can calculate coefficients independently using simple formulas.
• **Norm Simplicity**: The norm of any vector in terms of orthogonal components simplifies greatly. Each component can be handled individually without cross terms, thanks to orthogonality.
• **Reduction in Complexity**: If the orthogonal set is further normalized, meaning each basis vector is of unit length, it becomes an orthonormal basis, which is even more advantageous, providing direct readings of components with minimal computation.

These attributes make orthogonal bases of particular interest in both theoretical aspects and practical computations, especially in the realms of signal processing and numerical analysis.

Basics of Linear Transformation Properties

Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. They are the backbone of virtually all structural analysis in algebra. Here are some fundamental properties:

• **Additivity**: For any two vectors \(\mathbf{u}\) and \(\mathbf{v}\), a linear transformation \(T\) satisfies \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
• **Scalar Multiplication Compatibility**: For any scalar \(c\) and vector \(\mathbf{u}\), it holds that \( T(c\mathbf{u}) = cT(\mathbf{u}) \).
• **Representation by Matrices**: Linear transformations between finite-dimensional vector spaces can be fully characterized by matrices once a basis is chosen for each space.

These properties mean that linear transformations maintain the structure of the vector spaces they connect, allowing for predictable and organized manipulation of vectors. This predictability is what makes them so powerful and widely applicable in fields ranging from engineering to computer science.