Question

Let \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be given by $$ T(a, b)=(b-a, a+2 b) $$ Show that \(T\) is symmetric if the dot product is used in \(\mathbb{R}^{2}\) but that it is not symmetric if the following inner product is used: $$ \langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x} A \mathbf{y}^{T}, A=\left[\begin{array}{rr} 1 & -1 \\ -1 & 2 \end{array}\right] $$

Short answer

T is symmetric with the standard dot product but not symmetric with the custom inner product.

Step-by-step solution

Define Symmetry with Standard Dot Product

A linear transformation function \( T \) is symmetric with respect to the standard dot product if for any vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^2 \), we have \( \langle T(\mathbf{u}), \mathbf{v} \rangle = \langle \mathbf{u}, T(\mathbf{v}) \rangle \), where \( \langle \mathbf{x}, \mathbf{y} \rangle = \mathbf{x} \cdot \mathbf{y} \) is the standard dot product.

Check Symmetry with Standard Dot Product

For vectors \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \), find \( T(\mathbf{u}) = (u_2-u_1, u_1 + 2u_2) \) and \( T(\mathbf{v}) = (v_2-v_1, v_1 + 2v_2) \). Then compute: 1. \( \langle T(\mathbf{u}), \mathbf{v} \rangle = (u_2-u_1, u_1+2u_2) \cdot (v_1, v_2) = (u_2-u_1)v_1 + (u_1+2u_2)v_2 \) 2. \( \langle \mathbf{u}, T(\mathbf{v}) \rangle = (u_1, u_2) \cdot (v_2-v_1, v_1+2v_2) = u_1(v_2-v_1) + u_2(v_1+2v_2) \). Both expressions simplify to \( -u_1v_1 + u_2v_1 + u_1v_2 + 2u_2v_2 \), confirming symmetry.

Define Symmetry with Custom Inner Product

For the transformation to be symmetric under the custom inner product, it must hold that \( \langle T(\mathbf{u}), \mathbf{v} \rangle_A = \langle \mathbf{u}, T(\mathbf{v}) \rangle_A \), where \( \langle \mathbf{x}, \mathbf{y} \rangle_A = \mathbf{x} A \mathbf{y}^T \) with matrix \( A \).

Compute Inner Products with Matrix A

First, calculate each inner product: 1. \( \langle T(\mathbf{u}), \mathbf{v} \rangle_A = (u_2-u_1, u_1+2u_2) A (v_1, v_2)^T = (u_2-u_1, u_1+2u_2) \begin{pmatrix} v_1 - v_2, -v_1 + 2v_2 \end{pmatrix}^T = (u_2-u_1)(v_1 - v_2) + (u_1+2u_2)(-v_1 + 2v_2) \). 2. \( \langle \mathbf{u}, T(\mathbf{v}) \rangle_A = (u_1, u_2) A (v_2-v_1, v_1+2v_2)^T = (u_1, u_2) \begin{pmatrix} v_2 - v_1, -v_1 + 2v_2 \end{pmatrix}^T = u_1(v_2 - v_1) + u_2(-v_1 + 2v_2) \). Simplifying both expressions results in different terms, hence \( T \) is not symmetric with this inner product.

Key concepts

Symmetric Operations

A symmetric operation in the context of linear transformations is when a transformation acts the same on swapped variables, such that the order of the variables does not matter. This is frequently discussed using the standard dot product. Here, for a transformation to be symmetric under the dot product in a standard two-dimensional space, it must satisfy the condition that for any vectors \( \mathbf{u} \) and \( \mathbf{v} \) in the space, we have \( \langle T(\mathbf{u}), \mathbf{v} \rangle = \langle \mathbf{u}, T(\mathbf{v}) \rangle \). This means the operation behaves consistently regardless of the order in which vectors are applied.
When a linear transformation \( T \) operating on vector \( (a, b) \) in \( \mathbb{R}^2 \) is given by \( T(a, b) = (b-a, a+2b) \), we can verify its symmetry. By developing both expressions \( \langle T(\mathbf{u}), \mathbf{v} \rangle \) and \( \langle \mathbf{u}, T(\mathbf{v}) \rangle \) with standard dot products, if the results are equal, then \( T \) is symmetric under this product.

Dot Product

The dot product is a fundamental component of vector mathematics. It provides a scalar result from two vectors, showing an intuitive notion of their 'parallelness'. For vectors \( \mathbf{x} = (x_1, x_2) \) and \( \mathbf{y} = (y_1, y_2) \) in \( \mathbb{R}^2 \), the dot product is calculated as:\[\mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2\]
This measure has significant importance in verifying symmetric operations. In our original exercise, the dot product helps test whether the transformation \( T \) is symmetric under standard circumstances by applying the rule:

• Calculate \( T(\mathbf{u}) \cdot \mathbf{v} \)
• Calculate \( \mathbf{u} \cdot T(\mathbf{v}) \)

If both are equal, symmetry with respect to the dot product is confirmed.

Inner Product

The concept of inner products extends the idea of the dot product to involve matrices or more generalized vector spaces. An inner product is a way of multiplying two vectors to get a scalar, with certain mathematical properties such as being distributive, commutative, and positive definite.
In the context of the exercise, we are considering a custom inner product defined as:\[\langle \mathbf{x}, \mathbf{y} \rangle_A = \mathbf{x} A \mathbf{y}^T\]where matrix \( A \) is given. For the transformation to remain symmetric under this new definition, it should satisfy:\[\langle T(\mathbf{u}), \mathbf{v} \rangle_A = \langle \mathbf{u}, T(\mathbf{v}) \rangle_A\]By calculating these values with the given matrix \( A \), if the result varies, as is the case here, the transformation is not symmetric with this inner product choice.

Matrix Algebra

Matrix algebra provides the framework for handling operations like transformations on vectors. It allows for the representation of linear transformations in terms of matrices and coordinates, offering versatility in solution approaches involving vectors.
In exploring the original problem, understanding how matrices operate on vectors is crucial. The transformation \( T(a, b) \) can be viewed in terms of matrix multiplication when using the custom inner product. Here, the matrix \( A = \begin{bmatrix} 1 & -1 \ -1 & 2 \end{bmatrix} \) plays a vital role by transforming vectors through operations: it links the concept of applied transformation through rows and columns of a matrix with the element-wise operations on vector pairs.
By applying these matrix operations in various inner product settings, significant insights are gained into symmetrical behavior and transformations. Comprehending how to manipulate and interpret these operations is a strong foundation for advanced topics in linear algebra.