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Chapter 6: Problem 9

Find the matrix \([T]_{c+5}\) of the linear transformation \(T: V \rightarrow W\) with respect to the bases \(\mathcal{B}\) and \(\mathcal{C}\) of \(V\) and \(W\), respectively. Verify Theorem 6.26 for the vector \(\mathbf{v}\) by computing \(T(\mathbf{v})\) directly and using the theorem. $$\begin{array}{l} T: M_{22} \rightarrow M_{22} \text { defined by } T(A)=A^{T}, B=C= \\ {\left[E_{11}, E_{12}, E_{21}, E_{22}\right\\}, \mathbf{v}=A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]} \end{array}$$

Short Answer

Expert verified
Matrix \([T]_{c+5}\) is \(\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\), and \(T(\mathbf{v})\) verifies the theorem.

Step by step solution

01

Identifying the Transformation and Bases

The given linear transformation is \(T: M_{22} \to M_{22}\) defined by \(T(A) = A^T\). The canonical bases of \(M_{22}\) is \(\mathcal{B} = \mathcal{C} = \{E_{11}, E_{12}, E_{21}, E_{22}\}\), where \(E_{ij}\) is the matrix with 1 at the \((i,j)\)-location and 0 elsewhere. The vector \(\mathbf{v} = A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\).
02

Compute the Action of T on Basis Elements

Apply \(T\) to each basis element. \[ T(E_{11}) = E_{11}, \; T(E_{12}) = E_{21}, \; T(E_{21}) = E_{12}, \; T(E_{22}) = E_{22} \]
03

Express Outputs in Terms of Basis

Each transformed basis element is already expressed as a linear combination of the original basis: \[ T(E_{11}) = E_{11}, \; T(E_{12}) = E_{21}, \; T(E_{21}) = E_{12}, \; T(E_{22}) = E_{22}. \]
04

Construct the Matrix \([T]_{c+5}\)

The matrix \([T]_{c+5}\) is constructed by using the coefficients of the above expressions. Thus, \([T]_{c+5} = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}\).
05

Compute \(T(\mathbf{v})\) Directly

Directly compute \(T(\mathbf{v})\) using the definition of \(T\): \[ T\left(\begin{bmatrix} a & b \ c & d \end{bmatrix}\right) = \begin{bmatrix} a & c \ b & d \end{bmatrix}. \]
06

Verify Theorem 6.26 with Matrix Representation

To verify the theorem, compute \([\mathbf{v}]_{\mathcal{B}} = \begin{bmatrix} a \ b \ c \ d \end{bmatrix}\). Apply the transformation matrix \([T]_{c+5}\) to get: \[ [T(\mathbf{v})]_{\mathcal{C}} = [T]_{c+5} \cdot [\mathbf{v}]_{\mathcal{B}} = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} a \ b \ c \ d \end{bmatrix} = \begin{bmatrix} a \ c \ b \ d \end{bmatrix}. \]
07

Comparison of Results

The result from computing \(T(\mathbf{v})\) directly, \(\begin{bmatrix} a & c \ b & d \end{bmatrix}\), matches the coordinates result \([T(\mathbf{v})]_{\mathcal{C}} = \begin{bmatrix} a \ c \ b \ d \end{bmatrix}\), verifying Theorem 6.26.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Transposition
Matrix transposition is a fundamental operation in linear algebra that involves swapping the rows and columns of a matrix. Given a matrix \( A \), its transpose, denoted as \( A^T \), is constructed by taking each row of the original matrix and turning it into a column in the transposed matrix. This concept is crucial in various applications across mathematics and computer science.

  • For a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the transpose \( A^T \) is \( \begin{bmatrix} a & c \ b & d \end{bmatrix} \).
  • Matrix transposition is a linear operation, meaning \( (A + B)^T = A^T + B^T \) and \( (cA)^T = cA^T \) for any scalar \( c \).
  • Transposing a matrix twice will return the original matrix: \( (A^T)^T = A \).
Understanding matrix transpositions is essential for solving problems related to linear transformations since transpositions often represent changes in orientation or perspective.
Basis of a Vector Space
A basis of a vector space is a set of vectors that are linearly independent and span the vector space. This means each vector in the space can be expressed as a linear combination of the basis vectors, and no basis vector can be written as a combination of others. Finding a basis is essential as it provides a coordinate system for the vector space, allowing us to describe any vector uniquely.

  • For the space of \( 2 \times 2 \) matrices, a common basis is \( \{E_{11}, E_{12}, E_{21}, E_{22}\} \).
  • Each basis element \( E_{ij} \) is a matrix with 1 at position \( (i, j) \) and 0 elsewhere.
  • This set is used in the given exercise to express matrices and transformations.
Understanding the concept of a basis helps in decomposing complex vector transformations into simpler, more manageable components.
Matrix Representation
The matrix representation of a linear transformation involves expressing the transformation as a matrix that acts on a vector. This allows us to use matrix operations to compute transformations efficiently. The matrix representation is determined by how the transformation rearranges the basis vectors of the domain to form the basis vectors of the codomain.

In this context, the matrix \([T]_{c+5}\) represents the linear transformation \( T(A) = A^T \) with respect to the basis \( \{E_{11}, E_{12}, E_{21}, E_{22}\} \). Here's how it works:

  • The matrix \([T]_{c+5}\) is \( \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} \), indicating how \( T \) reorders the basis elements.
  • It permutes indices according to the transpose operation: switching \( E_{12} \) and \( E_{21} \).
  • This representation simplifies the computation of the transformation for any matrix \( A \) expressed in the basis \( \mathcal{B} \).
Matrix representation bridges abstract transformations with concrete calculations, making them amenable to algorithmic processing.
Linear Algebra Theorem Verification
Verifying a theorem in linear algebra typically involves proving that an algebraic statement holds for a specific example or situation. In this exercise, Theorem 6.26 is verified for the vector \( \mathbf{v} \) by comparing the results of two transformation approaches.

  • Firstly, calculate \( T(\mathbf{v}) \) directly from the transformation definition, yielding \( \begin{bmatrix} a & c \ b & d \end{bmatrix} \).
  • Secondly, convert \( \mathbf{v} \) into coordinates relative to the basis \( \mathcal{B} \), apply the transformation matrix \([T]_{c+5}\), and observe the output.
  • The agreement between direct computation and matrix representation results confirms the theorem's validity, illustrating the consistency of matrix transformations with theoretical expectations.
Such theorem verification exercises deepen understanding of both the theoretical and computational aspects of linear transformations, ensuring students can confidently apply abstract concepts to practical problems.

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Most popular questions from this chapter

Find the coordinate vector of \(p(x)=2-x+3 x^{2}\) with respect to the basis \(\mathcal{B}=\left\\{1+x, 1-x, x^{2}\right\\}\) of \(\mathscr{P}_{2}\)

Let \(\\{\mathbf{u}, \mathbf{v}, \mathbf{w}\\}\) be a linearly independent set of vectors in a vector space \(V\) (a) \(\mathrm{Is}\\{\mathbf{u}+\mathbf{v}, \mathbf{v}+\mathbf{w}, \mathbf{u}+\mathbf{w}\\}\) linearly independent? Either prove that it is or give a counterexample to show that it is not. (b) \(\mathrm{Is}\\{\mathbf{u}-\mathbf{v}, \mathbf{v}-\mathbf{w}, \mathbf{u}-\mathbf{w}\\}\) linearly independent? Either prove that it is or give a counterexample to show that it is not.

Show that \(T: \mathscr{P}_{n} \rightarrow \mathscr{P}_{n}\) defined by \(T(p(x))=p(x)+\) \(p^{\prime}(x)\) is an isomorphism.

Find the solution of the differential equation that satisfies the given boundary condition(s). $$f^{\prime \prime}-f^{\prime}-f=0, f(0)=0, f(1)=1$$

(a) Show that \(\mathscr{C}[0,1] \cong \mathscr{C}[2,3] .[\text { Hint: Define } T:\) \(\mathscr{C}[0,1] \rightarrow \mathscr{C}[2,3]\) by letting \(T(f)\) be the function whose value at \(x\) is \((T(f))(x)=f(x-2)\) for \(x\) in \([2,3] .]\) (b) Show that \(\mathscr{C}[0,1] \cong \mathscr{C}[a, a+1]\) for all \(a\).
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