Question

Let \(V\) be an inner product space. If \(U \subseteq V\) is a subspace, show that \(U\) is an inner product space using the same inner product.

Short answer

U is an inner product space since it inherits the properties of the inner product from V.

Step-by-step solution

Define Inner Product Space and Subspace

An inner product space is a vector space V along with an inner product \(\langle u, v \rangle\) for vectors \(u, v \in V\), which satisfies linearity, symmetry (conjugate symmetry for complex numbers), and positive-definiteness. A subspace \(U\) of \(V\) is a subset of \(V\) that is also a vector space under the operations defined in \(V\). We want to show that \(U\) is an inner product space with the same inner product inherited from \(V\).

Verify Closure of Inner Product in U

To use the inner product from \(V\) within \(U\), we first verify that \(\langle u, v \rangle\) is closed in \(U\). For any \(u, v \in U\), since \(U\) is a subspace, \(\langle u, v \rangle\) defined in \(V\) is still valid and remains within \(U\).

Check Linearity in the First Argument

For \(U\) to be an inner product space, the inner product must be linear. For any \(a, b \in \mathbb{R}\) (or \(\mathbb{C}\)) and \(u, v, w \in U\), the inner product satisfies: \(\langle au + bw, v \rangle = a\langle u, v \rangle + b\langle w, v \rangle\). This follows directly from the properties of the inner product in \(V\).

Confirm Symmetry

The inner product must be symmetric, where \(\langle u, v \rangle = \langle v, u \rangle\) for all \(u, v \in U\). This property is inherited directly from the inner product defined in \(V\).

Ensure Positive-Definiteness

Finally, check that the inner product is positive-definite: \(\langle u, u \rangle \geq 0\) and \(\langle u, u \rangle = 0\) if and only if \(u = 0\). This condition is satisfied in \(U\) because it's a subspace of \(V\), where this property holds.

Key concepts

Subspace

In linear algebra, a subspace is essentially a smaller space within a larger vector space that retains the structure and operations of the larger space. For a set to be a subspace, it must satisfy three crucial criteria:

• Contain the zero vector.
• Be closed under vector addition.
• Be closed under scalar multiplication.

In the context of inner product spaces, if we have a subspace \(U\) of \(V\), all vectors in \(U\) can be manipulated using the same operations and inner product defined in \(V\). This means that we do not need to redefine the inner product for \(U\), making it convenient to analyze subspaces with familiar rules.

Linearity

Linearity is a fundamental property of inner products, which helps in simplifying operations and equations in vector spaces. An inner product \(\langle u, v \rangle\) is said to be linear in its first argument if for any scalars \(a, b\) and vectors \(u, v, w \, \in V\):

• \(\langle au + bw, v \rangle = a\langle u, v \rangle + b\langle w, v \rangle\)

This property ensures that inner products behave predictably under addition and scalar multiplication. By preserving linearity in \(U\), the calculations remain consistent with those in the entire vector space \(V\). It automatically follows from the linearity in \(V\), making \(U\) an inner product space under the inherited operations.

Symmetry

Symmetry in the context of inner products refers to the interchangeability of input vectors without altering the result of the inner product. Formally, for any vectors \(u, v \in V\), the symmetry property is defined as:

• \(\langle u, v \rangle = \langle v, u \rangle\)

For complex vector spaces, this becomes conjugate symmetry:

• \(\langle u, v \rangle = \overline{\langle v, u \rangle}\)

This mirrors how inner product results are consistent regardless of the order of inputs. By extending this property to subspaces like \(U\), the calculations remain straightforward and rely on established properties from \(V\). In essence, it maintains the logic and orderliness that symmetry brings to mathematical operations.

Positive-Definiteness

Positive-definiteness is a key property of inner products that ensures a sense of distance or measure of magnitude between vectors. For a vector \(u \in V\), positive-definiteness is expressed as:

• \(\langle u, u \rangle \geq 0\) - The product is non-negative.
• \(\langle u, u \rangle = 0\) if and only if \(u = 0\) - The product is zero only for the zero vector.

This property effectively measures the 'length' or 'size' of a vector, which is crucial for geometry in vector spaces. When applied to the subspace \(U\), it means that the vectors within \(U\) can also be measured using the same standard. This allows us to work with subspaces as we would with the entire space \(V\), ensuring that \(U\) is a complete inner product space in its own right.