Question

\(V\) denotes a finite dimensional inner product space. If \(B\) and \(D\) are any orthonormal bases of \(V\), show that there is an isometry \(T: V \rightarrow V\) that carries \(B\) to \(D\)

Short answer

An isometry \( T : V \rightarrow V \) exists carrying \( B \) to \( D \), preserving inner products.

Step-by-step solution

Understanding Orthonormal Bases

An orthonormal basis for a vector space is a basis where all vectors are orthogonal to each other and have a unit norm (length of 1). This means if \( B = \{b_1, b_2, \ldots, b_n\} \) and \( D = \{d_1, d_2, \ldots, d_n\} \) are orthonormal bases for \( V \), each pair of vectors within each basis is orthogonal and each vector has norm 1.

Defining the Isometry

An isometry is a linear map \( T : V \rightarrow V \) that preserves inner products. Therefore, for any vectors \( u, v \in V \) and under the map \( T \), the condition \( \langle T(u), T(v) \rangle = \langle u, v \rangle \) must hold. To construct such a map, we need \( T \) to carry the basis \( B \) to the basis \( D \).

Constructing the Linear Map

Define the linear map \( T : V \rightarrow V \) by setting \( T(b_i) = d_i \) for each \( i \) from 1 to \( n \). Since both bases are orthonormal, this map is well-defined and uniquely extends to all of \( V \) as a linear transformation.

Verification of Isometry Condition

Verify that \( T \) preserves the inner product. Since \( T(b_i) = d_i \) and both bases are orthonormal, \( \langle T(b_i), T(b_j) \rangle = \langle d_i, d_j \rangle = \delta_{ij} \), which matches \( \langle b_i, b_j \rangle = \delta_{ij} \), where \( \delta_{ij} \) is the Kronecker delta.

Conclusion on Isometry

Since \( T \) preserves inner products for all basis vectors and extends linearly, this proves \( \langle T(u), T(v) \rangle = \langle u, v \rangle \) for all \( u, v \in V \). Therefore, \( T \) is an isometry that carries the basis \( B \) onto the basis \( D \).

Key concepts

Orthonormal Basis

An orthonormal basis in a vector space is a special kind of basis. It consists of vectors that are both orthogonal to each other and have a length, also known as norm, of one. Mathematically, if you have a set of vectors \( B = \{b_1, b_2, \ldots, b_n\} \), it forms an orthonormal basis if the inner product \( \langle b_i, b_j \rangle = 0 \) for all \( i eq j \) and \( \langle b_i, b_i \rangle = 1 \) for all \( i \).
Such a basis simplifies many calculations in vector spaces because it directly gives us coordinates of vectors, and any vector in the space can be easily represented as a combination of these basis vectors.

Orthonormal bases are incredibly useful in simplifying problems involving distances and angles within vector spaces, particularly in finite-dimensional inner product spaces like \( V \). It is also worth noting that any two orthonormal bases of a given space \( V \) can be connected via a linear mapping, known as an isometry, due to their structural properties.

Inner Product Space

An inner product space is a vector space equipped with an additional structure called the inner product. The inner product is a way of multiplying two vectors to produce a scalar. This structure allows us to define important geometric concepts such as length, distance, and angle within the vector space.
For vectors \( u \) and \( v \) in an inner product space \( V \), the inner product \( \langle u, v \rangle \) satisfies the following properties:

• Linearity: The inner product is linear in its arguments, meaning it distributes over vector addition and is compatible with scalar multiplication.
• Symmetry: The inner product is symmetric, so \( \langle u, v \rangle = \langle v, u \rangle \).
• Positive Definiteness: The inner product of a vector with itself is always positive and is zero if and only if the vector is the zero vector.

These properties make inner product spaces fundamental in functional analysis and important in any field that relies on geometric reasoning, such as quantum mechanics and signal processing.

Linear Map

A linear map, also known as a linear transformation, is a function between two vector spaces that preserves vector addition and scalar multiplication. Given vector spaces \( U \) and \( V \), a map \( T: U \rightarrow V \) is linear if for any vectors \( u, v \in U \) and any scalar \( c \), the following holds:

• \( T(u + v) = T(u) + T(v) \) (Additivity)
• \( T(cu) = cT(u) \) (Homogeneity)

Linear maps play a crucial role in various mathematical concepts and applications, particularly when they work as isometries. An isometry is a special type of linear map that preserves the inner product between vectors, meaning it maintains the angles and distances between vectors. In this context, if we need to map one orthonormal basis to another while preserving these distances and angles, the transformation \( T \) becomes an isometry.
By establishing \( T(b_i) = d_i \) where \( \{b_i\} \) and \( \{d_i\} \) are orthonormal bases, we ensure \( T \) defines an isometry from basis \( B \) to basis \( D \). This property is critical in ensuring that the mapping behaves consistently and predictably across the vector space.