Question

Let \(\langle,\rangle\) be an inner product on a vector space \(V\). Show that the corresponding distance function is translation invariant. That is, show that \(\mathrm{d}(\mathbf{v}, \mathbf{w})=\mathrm{d}(\mathbf{v}+\mathbf{u}, \mathbf{w}+\mathbf{u})\) for all \(\mathbf{v}, \mathbf{w},\) and \(\mathbf{u}\) in \(V\)

Short answer

The distance function is translation invariant because it remains unchanged when both vectors are translated by the same vector.

Step-by-step solution

Define the Distance Function

The distance function corresponding to the inner product is given by \( \mathrm{d}(\mathbf{v}, \mathbf{w}) = \sqrt{\langle \mathbf{v} - \mathbf{w}, \mathbf{v} - \mathbf{w} \rangle} \). This measures the 'distance' between two vectors \( \mathbf{v} \) and \( \mathbf{w} \) in the vector space \( V \).

Translate the Distance Function

We want to show that translating both vectors \( \mathbf{v} \) and \( \mathbf{w} \) by a vector \( \mathbf{u} \) results in the same distance. This means showing \( \mathrm{d}(\mathbf{v}+\mathbf{u}, \mathbf{w}+\mathbf{u}) = \mathrm{d}(\mathbf{v}, \mathbf{w}) \).

Calculate the Translated Distance Function

Substitute \( \mathbf{v} + \mathbf{u} \) and \( \mathbf{w} + \mathbf{u} \) into the distance function: \( \mathrm{d}(\mathbf{v}+\mathbf{u}, \mathbf{w}+\mathbf{u}) = \sqrt{\langle (\mathbf{v} + \mathbf{u}) - (\mathbf{w} + \mathbf{u}), (\mathbf{v} + \mathbf{u}) - (\mathbf{w} + \mathbf{u}) \rangle} \).

Simplify the Expression

Notice that \((\mathbf{v} + \mathbf{u}) - (\mathbf{w} + \mathbf{u}) = \mathbf{v} - \mathbf{w}\). So we have \( \mathrm{d}(\mathbf{v}+\mathbf{u}, \mathbf{w}+\mathbf{u}) = \sqrt{\langle \mathbf{v} - \mathbf{w}, \mathbf{v} - \mathbf{w} \rangle} \).

Conclude Translation Invariance

The expression \( \sqrt{\langle \mathbf{v} - \mathbf{w}, \mathbf{v} - \mathbf{w} \rangle} \) on the right side is the same as the original distance function \( \mathrm{d}(\mathbf{v}, \mathbf{w}) \), thus proving translation invariance: \( \mathrm{d}(\mathbf{v}, \mathbf{w}) = \mathrm{d}(\mathbf{v}+\mathbf{u}, \mathbf{w}+\mathbf{u}) \).

Key concepts

Translation Invariance

Translation invariance is a key property of distance functions associated with inner product spaces. In simple terms, translation invariance means that moving (or translating) both points in space by the same amount does not change the distance between them.

To understand why this is true in the context of inner product spaces, consider the distance function, which is defined as \( \mathrm{d}(\mathbf{v}, \mathbf{w}) = \sqrt{\langle \mathbf{v} - \mathbf{w}, \mathbf{v} - \mathbf{w} \rangle} \). When both vectors \(\mathbf{v}\) and \(\mathbf{w}\) are translated by a vector \(\mathbf{u}\), they become \(\mathbf{v} + \mathbf{u}\) and \(\mathbf{w} + \mathbf{u}\), respectively.

The translated distance is \( \sqrt{\langle (\mathbf{v} + \mathbf{u}) - (\mathbf{w} + \mathbf{u}), (\mathbf{v} + \mathbf{u}) - (\mathbf{w} + \mathbf{u}) \rangle} \). Note the cancellation: when you subtract, the \(\mathbf{u}\)'s disappear, resulting in \(\mathbf{v} - \mathbf{w}\). The distance formula thus remains unchanged, demonstrating translation invariance.

Vector Spaces

Vector spaces are fundamental structures in mathematics that provide the setting for inner products and distance functions. A vector space is essentially a collection of vectors, where vectors can be added together and multiplied by scalars, all while obeying certain rules (axioms).

Here's what you need to know about vector spaces:

• The sum of any two vectors in the space is also in the space, and this operation is commutative and associative.
• Every vector space includes a zero vector, which is the additive identity.
• Each vector has an inverse, meaning for any vector \(\mathbf{v}\), there exists a vector \(-\mathbf{v}\) such that \(\mathbf{v} + (-\mathbf{v}) = \mathbf{0}\).
• Scalar multiplication is consistent, meaning multiplying a vector by a scalar does not change the vector’s nature.

These properties make vector spaces understandable and manageable, allowing for operations like inner products and translations to be well-defined and meaningful.

Distance Function

Distance functions in the context of inner product spaces allow us to measure the 'distance' or separation between two vectors. This measurement is not physical distance in a literal sense, but rather a metric that enables specific comparisons and analyses within vector spaces.

The distance function derived from an inner product is given by \( \mathrm{d}(\mathbf{v}, \mathbf{w}) = \sqrt{\langle \mathbf{v} - \mathbf{w}, \mathbf{v} - \mathbf{w} \rangle} \), where \(\langle \cdot , \cdot \rangle\) denotes the inner product. Key things to remember about this function are:

• It is always non-negative as the square root of a squared term implies actual numbers.
• It is zero if and only if the two vectors are identical, meaning \(\mathbf{v} = \mathbf{w}\).
• It satisfies the triangular inequality, an essential property for any valid distance metric.

Understanding the distance function is crucial in exploring concepts such as orthogonality, projections, and various formulations of vector equations within inner product spaces.