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In this exercise, we show that every inner product ⟨·, ·⟩on...

In this exercise, we show that every inner product $\langle\cdot, \cdot\rangle$ on $\mathbb{R}^n$ can be reduced to the dot product when expressed in a suitably adapted basis. (a) Specifically, prove that there exists a basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $\mathbb{R}^n$ such that $\langle\mathbf{x}, \mathbf{y}\rangle=\sum_{i=1}^n c_i d_i=\mathbf{c} \cdot \mathbf{d}$, where $\mathbf{c}=\left(c_1, c_2, \ldots, c_n\right)^T$ are the coordinates of $\mathbf{x}$ and $\mathbf{d}=\left(d_1, d_2, \ldots, d_n\right)^T$ those of $\mathbf{y}$ with respect to the basis. Is the basis uniquely determined? (b) Find bases that reduce the following inner products to the dot product on $\mathbb{R}^2$ : (i) $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 w_1+3 v_2 w_2$, (ii) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1-v_1 w_2-v_2 w_1+3 v_2 w_2$.

In this exercise, we show that every inner product $\langle\cdot, \cdot\rangle$ on $\mathbb{R}^n$ can be reduced to the dot product when expressed in a suitably adapted basis. (a) Specifically, prove that there exists a basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $\mathbb{R}^n$ such that $\langle\mathbf{x}, \mathbf{y}\rangle=\sum_{i=1}^n c_i d_i=\mathbf{c} \cdot \mathbf{d}$, where $\mathbf{c}=\left(c_1, c_2, \ldots, c_n\right)^T$ are the coordinates of $\mathbf{x}$ and $\mathbf{d}=\left(d_1, d_2, \ldots, d_n\right)^T$ those of $\mathbf{y}$ with respect to the basis. Is the basis uniquely determined? (b) Find bases that reduce the following inner products to the dot product on $\mathbb{R}^2$ :
(i) $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 w_1+3 v_2 w_2$,
(ii) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1-v_1 w_2-v_2 w_1+3 v_2 w_2$.

Key Concepts

Inner Product Space

An inner product space is a vector space equipped with an inner product, which is a function that assigns a scalar to each pair of vectors and satisfies properties like linearity, symmetry, and positive definiteness. This structure allows the notion of angles and lengths to be introduced in the vector space, leading to the concepts of orthogonality and norms, and ultimately providing the groundwork for geometric interpretations in abstract settings.

Basis Transformation

A basis transformation involves changing the coordinate system in a vector space by selecting a different basis. This change of basis can simplify problems by representing vectors and linear mappings in a form where certain properties, such as the form of an inner product, are easier to handle. In the context of inner product spaces, it is often possible to choose a basis such that the inner product takes on a canonical or simpler form, like the standard dot product.

Symmetric Bilinear Forms and Diagonalization

A symmetric bilinear form is an expression that takes two vectors and produces a scalar in a manner that is linear in each argument and symmetric with respect to their interchange. Representing this form by a symmetric matrix allows us to apply diagonalization techniques. The spectral theorem guarantees that any real symmetric matrix can be diagonalized by an orthogonal transformation, meaning there exists a basis in which the inner product is expressed as a sum of scaled component-wise products, often reducing to a dot product when appropriately normalized.

Uniqueness of Diagonalizing Bases

While it is always possible to find a basis that diagonalizes a symmetric bilinear form or inner product, this basis is generally not unique. The non-uniqueness arises due to the freedom in choosing the order of the basis vectors and their signs, as well as potential additional degrees of freedom when eigenvalues have multiplicity. This inherent non-uniqueness is a common theme in linear algebra whenever eigen-decompositions or diagonalizations are performed.

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