Question

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

Short answer

a. Yes, symmetric with dot product; b. No, not symmetric with matrix-based inner product.

Step-by-step solution

Recall the Definition of Symmetry

A transformation \(T\) is symmetric with respect to the dot product if for all vectors \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^2\), \(\langle T(\mathbf{x}), \mathbf{y} \rangle = \langle \mathbf{x}, T(\mathbf{y}) \rangle\). Using the standard dot product, this means that for any vectors \((a,b)\) and \((c,d)\), we need to check if \((2a+b, a-b) \cdot (c,d) = (2c+d, c-d) \cdot (a,b)\).

Verify Symmetry with Dot Product

Calculate \(\langle T(a,b), (c,d) \rangle = (2a+b, a-b) \cdot (c,d) = (2a+b)c + (a-b)d\). Compute the other side of the symmetry condition: \(\langle (a,b), T(c,d) \rangle = (a,b) \cdot (2c+d, c-d) = a(2c+d) + b(c-d)\). Simplifying, both expressions are equal to \(2ac + bd + bc - ad\). Since both sides are equal for any \(a,b,c,d\), \(T\) is symmetric with the dot product.

Recall Alternate Inner Product Definition

The alternative inner product is defined as \(\langle \mathbf{x}, \mathbf{y} \rangle = \mathbf{x} A \mathbf{y}^T\), where \(A\) is the given matrix. For vectors \(\mathbf{x} = (a,b)\) and \(\mathbf{y} = (c,d)\), calculate using \(A = \begin{pmatrix} 1 & 1 \ 1 & 2 \end{pmatrix}\).

Check Non-Symmetry with Alternate Product

First, compute \(\langle T(a,b), (c,d) \rangle\): Compute \(T(a, b) A (c,d)^T = (2a+b, a-b) \begin{pmatrix} 1 & 1 \ 1 & 2 \end{pmatrix} \begin{pmatrix} c \ d \end{pmatrix}\), giving components as \((2a+b)c + (a-b)(c+2d)\). Separately, compute \(\langle (a,b), T(c,d) \rangle = (a,b) \begin{pmatrix} 1 & 1 \ 1 & 2 \end{pmatrix} (2c+d, c-d)^T\), yielding a different expression: \(a(2c+d) + b(3c-d)\). These two expressions are not equal in general, so \(T\) is not symmetric with this inner product.

Key concepts

Symmetric Transformations

A symmetric transformation is one that respects the structure of the space it transforms, especially in terms of inner product relationships. For a transformation like \(T: \mathbb{R}^{2} \to \mathbb{R}^{2}\) to be symmetric with respect to the dot product, it must satisfy the condition \( \langle T(\mathbf{x}), \mathbf{y} \rangle = \langle \mathbf{x}, T(\mathbf{y}) \rangle\) for all vectors \(\mathbf{x}, \mathbf{y} \) in the space. This means when you apply \(T\) to any vector \(\mathbf{x}\) and take the dot product with another vector \(\mathbf{y}\), the result should be the same as applying \(T\) to \(\mathbf{y}\) and then taking the dot product with \(\mathbf{x}\).
Applying this definition helps in understanding how transformations behave and ensures that the transformation preserves the symmetrical properties characterized by the dot product. In simpler terms, this condition ensures the angle and length relationships between vectors are preserved under the transformation.

Dot Product

The dot product, also known as the scalar product, is a fundamental algebraic operation for vectors. It calculates a single number (a scalar) from two equal-length sequences of numbers (usually coordinate vectors). The standard dot product for two vectors \((a, b)\) and \((c, d)\) is calculated as:\[ a \cdot c + b \cdot d \]This operation has several important properties:

• Commutative: \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\)
• Distributive over addition: \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}\)
• Scalar multiplication compatibility: \((k\mathbf{a}) \cdot \mathbf{b} = k(\mathbf{a} \cdot \mathbf{b})\)

The dot product is key in determining the angle between two vectors, with a dot product of zero implying perpendicular vectors.

Matrix Multiplication

Matrix multiplication is an operation that takes a pair of matrices and produces another matrix. In the context of transformations like the one studied in the exercise, this operation is crucial for understanding how transformations work when different bases or inner products are involved. A matrix \(A\) can represent the transformation \(T\) such that \(T(\mathbf{x}) = \mathbf{x}A\), where \(\mathbf{x}\) is a row or column vector depending on the context.
When associating matrices with transformations, you multiply matrices in the order of the transformation sequence:
\[ (AB)C = A(BC) \]
This operation is not commutative, meaning \(AB eq BA\) in general. Understanding matrix multiplication helps in finding out how different transformations stack together and affect vectors in a space, making it crucial for topics like linear transformations and other advanced linear algebra concepts.

Inner Product Spaces

Inner product spaces are vector spaces equipped with an additional feature known as an inner product. This additional structure allows for the concepts of length and angles to be extended to more abstract vector spaces. The most common example of an inner product is the dot product. An inner product must satisfy several properties:

• Conjugate symmetry: \(\langle \mathbf{x}, \mathbf{y} \rangle = \overline{\langle \mathbf{y}, \mathbf{x} \rangle}\)
• Linearity in the first argument: \(\langle a\mathbf{x} + b\mathbf{y}, \mathbf{z} \rangle = a\langle \mathbf{x}, \mathbf{z} \rangle + b\langle \mathbf{y}, \mathbf{z} \rangle \)
• Positive-definiteness: \(\langle \mathbf{x}, \mathbf{x} \rangle \geq 0\) with equality if and only if \(\mathbf{x} = 0\)

In the exercise, the focus is on an alternative inner product defined by a specific matrix \(A\), altering the usual structure of the space. This shift in inner product reflects in the behavior of transformations like \(T\), showing how essential inner product spaces are in determining symmetry properties.