Question

In each case, find the matrix of the linear transformation \(T: V \rightarrow W\) corresponding to the bases \(B\) and \(D\) of \(V\) and \(W\), respectively. a. \(T: \mathbf{M}_{22} \rightarrow \mathbb{R}, T(A)=\operatorname{tr} A\) \(\begin{aligned} B=&\left\\{\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right\\}, \\ D=\\{1\\} \end{aligned}\) b. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}, T(A)=A^{T}\) \(\left.\begin{array}{rl}B & =D \\ & =\left\\{\left[\begin{array}{ll}1 & 0 \\\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right\\}\end{array}\right\\}\) c. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{3}, T[p(x)]=x p(x) ; B=\left\\{1, x, x^{2}\right\\}\) and \(\quad D=\left\\{1, x, x^{2}, x^{3}\right\\}\) d. \(\quad T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}, T[p(x)]=p(x+1)\) \(\quad B=D=\left\\{1, x, x^{2}\right\\}\)

Short answer

a: [1, 0, 0, 1]^T; b: Identity matrix; c: Transformation matrix as given; d: Transformation matrix as given.

Step-by-step solution

Identify the Matrix for Transformation A (Part a)

For part a, the transformation is defined as multiplying a 2x2 matrix by its trace, which is a linear transformation from \(\mathbf{M}_{22}\) to \(\mathbb{R}\). The trace, \(\operatorname{tr} A\), of a 2x2 matrix \(A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right]\) is \(a + d\). We recognize that there are four basis matrices involved for \(V\). By exporting the trace function over each of the matrices in basis \(B\), we find out that the trace of \(B_1\) contributes 1, \(B_2\) and \(B_3\) contribute 0, and \(B_4\) contributes 1. Therefore, the matrix representation of \(T\) with respect to these bases is \([1, 0, 0, 1]^T\).

Identify the Matrix for Transformation B (Part b)

In part b, the linear transformation \(T\) is the transpose of matrix \(A\). The bases for \(V\) and \(W\) are identical and consist of the 2x2 matrices. Calculating the image of each basis element's transpose, we determine their columns in the matrix \(T\). Each matrix remains effectively a combination of permutated basis elements, resulting in the identity matrix as the transformation matrix: \(T = I_4\).

Identify the Matrix for Transformation C (Part c)

For part c, the transformation is taking a degree-2 polynomial and multiplying by \(x\), resulting in degree-3 polynomials. Consider \(p(x) = c_0 + c_1 x + c_2 x^2\); \(T[p(x)] = xp(x) = c_0x + c_1x^2 + c_2x^3\). By evaluating each basis polynomial in \(B\) under \(T\), we can express the result in terms of \(D = \{1, x, x^2, x^3\}\) as follows: - \(T[1]=x\)- \(T[x]=x^2\)- \(T[x^2]=x^3\).The resultant matrix of \(T\) relative to these bases is:\[\begin{bmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\].

Identify the Matrix for Transformation D (Part d)

In part d, the transformation \(T\) shifts the polynomial by 1, making it such that \(p(x)\) maps to \(p(x+1)\). Let's express \(p(x) = a + bx + cx^2\); then \(p(x+1) = a + b(x+1) + c(x+1)^2\). Simplifying and calculating with respect to \(B = D = \{1, x, x^2\}\) yields the transformed basis as:- \(T[1] = 1\)- \(T[x] = 1 + x\)- \(T[x^2] = 1 + 2x + x^2\).Thus, the matrix representation of \(T\) based on these basis sets is:\[\begin{bmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 2 & 1 \end{bmatrix}\].

Key concepts

Matrix Representation

In linear algebra, the concept of matrix representation of a linear transformation is crucial. Essentially, it allows us to express a transformation using matrices, which simplifies computations and visualizations. For instance, if we have a transformation \( T: V \to W \), where \( V \) and \( W \) are vector spaces, this transformation can be represented as a matrix \( A \) relative to specified bases of \( V \) and \( W \).
This representation depends on choosing appropriate bases for both the vector spaces. Once the bases are chosen, each element of the transformation can be written in terms of these basis vectors. The coefficients then form the columns of our matrix representation. This is helpful for applying the transformation to vectors in the vector space since we only need to perform matrix multiplication. The matrix representation simplifies the computation of the transformation, especially when dealing with compositions of transformations or their inverses.

• Finding a matrix representation involves determining how each basis vector from \( V \) maps to \( W \).
• It also provides insights into properties like rank and nullity of the linear transformation.

Polynomial Transformation

Polynomial transformations involve polynomials transitioning from one polynomial space to another via a specific operation, such as multiplication or shifting. For example, one transformation can map polynomials from the space of degree 2 to the space of degree 3 by multiplying them by \( x \). Consider the transformation \( T : \mathbf{P}_2 \to \mathbf{P}_3 \), where \( T[p(x)]=x p(x) \).
This means each polynomial \( p(x) \) in \( \mathbf{P}_2 \) is multiplied by \( x \) and results in a polynomial in \( \mathbf{P}_3 \). When evaluating how this transformation affects basis vectors such as \( \{1, x, x^2\} \) from \( \mathbf{P}_2 \):

• \( T[1] = x \)
• \( T[x] = x^2 \)
• \( T[x^2] = x^3 \)

These transformations can be expressed as matrix operations on vector representations of polynomials. Understanding polynomial transformations allows us to comprehend how polynomials can be manipulated through algebraic operations, expanding their applications from simple expression to complex functions.

Transpose of a Matrix

The transpose of a matrix is a fundamental operation in linear algebra, denoted \( A^T \) for any matrix \( A \). The operation involves flipping a matrix over its diagonal, effectively interchanging the row and column indices of each element. For an element \( a_{ij} \) in matrix \( A \), its position is swapped to \( a_{ji} \) in \( A^T \).
To illustrate, if you have a transformation \( T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} \) where \( T(A)=A^{T} \), it means the transformation takes each matrix and flips it. When performing this operation on a set of basis matrices, the matrix representation of \( T \) becomes the identity matrix because each basis matrix, after transposing, remains essentially a permutated version.

• The transpose operation is simple but has many profound implications in areas such as physics and computer graphics.
• It helps in solving equations, understanding symmetry properties, and is essential in defining other operations like the determinant and adjoint.

Basis and Dimension

In linear algebra, a basis of a vector space is a set of linearly independent vectors that span the entire space. Having a basis simplifies the description of every vector in the space as a linear combination of the basis vectors. For instance, to explore transformations, we first need to have clear basis sets for the involved vector spaces.
The dimension of a vector space is simply the number of vectors in a basis of the space. This provides a measure of the 'size' or 'capacity' of the vector space to accommodate vectors. For example, if you are dealing with matrices \( \mathbf{M}_{22} \), the dimension is 4, as there are 4 independent basis matrices.

• Choosing a basis is crucial; different bases can simplify problems in various ways, depending on the situation.
• The concept of dimension helps in understanding vector spaces and in calculating the matrix representation of transformations.

Using basis and dimension correctly can make it easier to perform operations with vectors, understand their relationships, and solve equations involving linear transformations.