Question

In each case, show that \(T\) is symmetric by calculating \(M_{B}(T)\) for some orthonormal basis \(B\). a. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) \(T(a, b, c)=(a-2 b,-2 a+2 b+2 c, 2 b-c) ;\) dot prod- uct b. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) $$ T\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{cc} c-a & d-b \\ a+2 c & b+2 d \end{array}\right] $$ inner product: $$ \left\langle\left[\begin{array}{cc} x & y \\ z & w \end{array}\right],\left[\begin{array}{cc} x^{\prime} & y^{\prime} \\ z^{\prime} & w^{\prime} \end{array}\right]\right\rangle=x x^{\prime}+y y^{\prime}+z z^{\prime}+w w^{\prime} $$ c. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) $$ T\left(a+b x+c x^{2}\right)=(b+c)+(a+c) x+(a+b) x^{2} $$ inner product: $$ \left\langle a+b x+c x^{2}, a^{\prime}+b^{\prime} x+c^{\prime} x^{2}\right\rangle=a a^{\prime}+b b^{\prime}+c c^{\prime} $$

Short answer

a. Symmetric; b. Not symmetric; c. Symmetric.

Step-by-step solution

Select Orthonormal Basis for Part a

For the transformation \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3\), we select the standard orthonormal basis \(B = \{e_1, e_2, e_3\}\) for \(\mathbb{R}^3\), where \(e_1 = (1, 0, 0)\), \(e_2 = (0, 1, 0)\), and \(e_3 = (0, 0, 1)\).

Compute Matrix Representation for Part a

Calculate the matrix representation \(M_B(T)\) by applying \(T\) to each basis vector:- \(T(e_1) = T(1, 0, 0) = (1, -2, 0)\)- \(T(e_2) = T(0, 1, 0) = (-2, 2, 2)\)- \(T(e_3) = T(0, 0, 1) = (0, 2, -1)\)Thus, the matrix \(M_B(T)\) is:\[\begin{bmatrix} 1 & -2 & 0 \-2 & 2 & 2 \0 & 2 & -1 \end{bmatrix}\]

Check Symmetry for Part a

The matrix \(M_B(T)\) is symmetric if \(M_B(T) = M_B(T)^T\). Here, the transpose of:\[\begin{bmatrix} 1 & -2 & 0 \-2 & 2 & 2 \0 & 2 & -1 \end{bmatrix}\]is:\[\begin{bmatrix} 1 & -2 & 0 \-2 & 2 & 2 \0 & 2 & -1 \end{bmatrix}\]Both are equal, hence \(T\) is symmetric.

Select Orthonormal Basis for Part b

For the transformation \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\), select the orthonormal basis for \(2 \times 2\) matrices as:\(B = \left\{\begin{bmatrix}1 & 0 \ 0 & 0 \end{bmatrix}, \begin{bmatrix}0 & 1 \ 0 & 0 \end{bmatrix}, \begin{bmatrix}0 & 0 \ 1 & 0 \end{bmatrix}, \begin{bmatrix}0 & 0 \ 0 & 1 \end{bmatrix}\right\}\).

Compute Matrix Representation for Part b

Apply \(T\) to each basis element:- \(T\left(\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\right) = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}\)- \(T\left(\begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\right) = \begin{bmatrix} 0 & -1 \ 0 & 2 \end{bmatrix}\)- \(T\left(\begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}\right) = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\)- \(T\left(\begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix}\right) = \begin{bmatrix} 0 & 0 \ 2 & 1 \end{bmatrix}\)Thus, the matrix \(M_B(T)\) is:\[\begin{bmatrix}0 & 0 & 1 & 0 \0 & -1 & 0 & 2 \1 & 0 & 0 & 0 \0 & 2 & 0 & 1 \end{bmatrix}\]

Check Symmetry for Part b

Examine the symmetry of \(M_B(T)\):\[\begin{bmatrix}0 & 0 & 1 & 0 \0 & -1 & 0 & 2 \1 & 0 & 0 & 0 \0 & 2 & 0 & 1 \end{bmatrix}\]Comparing this with its transpose, \(M_B(T)\) is not equal to \(M_B(T)^T\). Thus, \(T\) is not symmetric.

Select Orthonormal Basis for Part c

For \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\), use the basis \(B = \{1, x, x^2\}\), which is orthonormal with respect to the given inner product.

Compute Matrix Representation for Part c

Calculate \(M_B(T)\) by applying \(T\) to each basis vector:- \(T(1) = 0 + 1x + 1x^2 = x + x^2\)- \(T(x) = 1 + 1x + 0x^2 = 1 + x\)- \(T(x^2) = 1 + 0x + 1x^2 = 1 + x^2\)Thus, the matrix \(M_B(T)\) is:\[\begin{bmatrix} 0 & 1 & 1 \1 & 1 & 0 \1 & 0 & 1 \end{bmatrix}\]

Check Symmetry for Part c

Verify symmetry:\[\begin{bmatrix} 0 & 1 & 1 \1 & 1 & 0 \1 & 0 & 1 \end{bmatrix}\]Comparing with its transpose, they are equal. Therefore, \(T\) is symmetric.

Key concepts

Orthonormal Basis

An orthonormal basis is a set of vectors that are both orthogonal and normalized. In simple terms, orthogonal means each pair of different vectors in this set is perpendicular to each other (their inner product is zero), and normalized means each vector has a length (norm) of 1. This concept is crucial when working with symmetric matrices and linear transformations because it simplifies computations related to matrix operations and transformations.

The orthonormal basis makes calculations in linear algebra more efficient. For example, when we have a function managing 3D vectors, like in the original exercise with the transformation from \(\mathbb{R}^3 \to \mathbb{R}^3\), using the standard orthonormal basis of \(\{e_1, e_2, e_3\}\) simplifies the process of finding out how the transformation affects each direction in our space. The basis vectors \(e_1 = (1, 0, 0)\), \(e_2 = (0, 1, 0)\), and \(e_3 = (0, 0, 1)\) are perfect examples of an orthonormal basis.- **Orthogonal**: Vectors are perpendicular.- **Normalized**: Each vector has a magnitude of 1.- **Efficiency**: Simplifies calculations and matrix operations.

Matrix Representation

Matrix representation refers to expressing a linear transformation as a matrix. This is a powerful technique because it allows complex operations to be carried out as simple matrix multiplications. The matrix representation of a transformation depends on the basis you choose to use.

In the example of the transformation \(T: \mathbb{R}^3 \to \mathbb{R}^3\), applying \(T\) to each basis vector in the orthonormal basis \(B = \{e_1, e_2, e_3\}\) results in new vectors, which are the columns of the matrix representation \(M_B(T)\). The matrix keeps all the transformation properties succinctly represented, serving as a blueprint for transforming any vector in the space.
- **Matrix**: A two-dimensional array of numbers representing linear transformations.- **Columns**: Each column represents the transformation applied to each basis vector.- **Efficiency**: Enables fast and compact transformation operations.

Inner Product

The inner product (also known as a dot product) is a fundamental operation used to measure the angle and length relationships between vectors. In mathematics, the inner product of two vectors is a single number that captures the idea of multiplying them and summing their components. This product plays a vital role in defining orthonormality, understanding geometry in vector spaces, and working with transformations.

In linear transformations, the inner product helps define the space's geometry, as seen with transformations involving vectors. For matrix transformations like those in \(\mathbf{M}_{22}\), the inner product ensures the manipulation respects the structure of matrices and their space, facilitating the development of symmetries in transformations.
- **Measurement**: Determines orthogonality and magnitude relationships.- **Application**: Essential in defining symmetric transformations and matrices.- **Transformation**: Guides how transformations 'angle' and 'stretch' vectors in their space.

Linear Transformations

Linear transformations are mappings between vector spaces that preserve vector addition and scalar multiplication. Simply put, they are functions that change a vector but maintain the structure of addition and scaling.

A linear transformation can be represented by a matrix, linking the theory of abstract vector spaces with tangible calculations. In symmetric matrices, linear transformations reflect this symmetry; they remain unchanged when flipped along their diagonal. This property is observed when calculating the matrix representations for transformations like \(T: \mathbf{P}_{2} \to \mathbf{P}_{2}\) where symmetry was verified by comparing the matrix to its transpose.
- **Structure-Preserving**: Maintains vector operations (addition and scaling).- **Matrix Representation**: Transforms abstract ideas into computations.- **Symmetry**: In symmetric cases, the transformation matrix equals its transpose.