/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 The space of all 2 by 2 matrices... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 15

The space of all 2 by 2 matrices has the four basis "vectors" $$ \left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right] $$ For the linear transformation of transposing, find its matrix \(A\) with respect to this basis. Why is \(A^{2}=1 ?\)

Short Answer

Expert verified
\( A \) is \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \), and \( A^2 = I \). This is because the transpose of a transpose returns the original matrix.

Step by step solution

01

Understand the Transformation

The transformation is transposing a matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the transpose is \( \begin{bmatrix} a & c \ b & d \end{bmatrix} \).
02

Apply Transformation to Basis Matrices

Apply the transpose operation to each of the basis matrices:1. For \( \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \), the transpose is itself, \( \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \).2. For \( \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \), the transpose is \( \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \).3. For \( \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \), the transpose is \( \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \).4. For \( \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \), the transpose is itself, \( \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \).
03

Express Transpose Results in Basis Form

Express each of the transposed matrices in terms of the basis. We find:1. \( \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} = 1 \cdot \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \).2. \( \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} = 0 \cdot \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} + 1 \cdot \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \).3. \( \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} = 0 \cdot \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} + 1 \cdot \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \).4. \( \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} = 0 \cdot \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} + 0 \cdot \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} + 1 \cdot \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \).
04

Construct the Transformation Matrix A

The transformation matrix \( A \) is constructed using the coefficients from each expression formed in the previous step:\[ A = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} \].
05

Verify \( A^2 = I \)

To show \( A^2 = I \), compute \( A^2 \):\[ A^2 = A \cdot A = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} = I \].This is the identity matrix, so indeed \( A^2 = I \).

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Key Concepts

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

Matrix Transpose
Understanding the concept of matrix transpose is relatively straightforward. The transpose of any matrix is obtained by flipping it over its diagonal. This means that the first row becomes the first column, the second row becomes the second column, and so on.

Let's consider a 2x2 matrix for simplicity:
  • Given a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), its transpose is \( \begin{bmatrix} a & c \ b & d \end{bmatrix} \).
Matrix transposing is a linear transformation, which means it follows certain linearity properties such as scaling and addition. When applied to basis matrices, as in our exercise, we see this operation effect more clearly. Non-diagonal elements switch their positions, while diagonal elements remain unchanged. For example, transposing doesn't alter the identity matrix.
Basis Matrices
Basis matrices are fundamental building blocks for vector spaces, much like how individual numbers make up all numerical expressions. In the context of 2x2 matrices, a basis is a set of matrices that can be combined to express any 2x2 matrix.

The basis matrices for the space of all 2x2 matrices in our case are:
  • \( \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \)
  • \( \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \)
  • \( \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \)
  • \( \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \)
These matrices serve as a foundation, allowing any 2x2 matrix to be expressed as a linear combination of them. This is crucial in understanding the matrix representation of linear transformations, like the transpose, since the transformation can be simplified to working with this minimal set of matrices. Each transformation results in another set of basis matrices, retaining their form and interchanging elements according to the transformation's nature.
Identity Matrix
The identity matrix is a special type of square matrix with ones on the diagonal and zeros elsewhere. It is equivalent to the number '1' in multiplication, serving as the matrix counterpart of multiplying by one 鈥� meaning it doesn't change the other matrix when multiplied together.

The size of an identity matrix, denoted as \( I \), aligns with the dimensions of the matrices you are working with. So an identity matrix for a 2x2 system is:
  • \[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]
This matrix is significant in verifying transformations. In the transformation matrix \( A \) constructed for a transposing operation, when \( A \) is squared \( (A^2) \), it results in the identity matrix. This confirms the property that doing a transpose operation twice results in the original matrix 鈥� just as multiplying by '1' gives back the original number.
Matrix Representation
Understanding matrix representation involves expressing a linear transformation in terms of matrices. This simplifies complex problems into manageable computations. For our exercise, we explored the transformation of transposing matrices with respect to a given basis.

The core idea is to determine how the transformation affects each basis matrix, then use these changes to construct a new matrix, termed as the transformation matrix. In this exercise, applying the transpose operation to each basis matrix showed us the coefficients needed to express each result as a combination of the basis.

Consequently, we determined that:
  • \[ A = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} \]
This matrix \( A \) represents a transformation that, when applied to a matrix expressed in terms of the basis matrices, will transpose it. Successfully studying any transformation involves understanding its basis, expression in matrix form, and verification by identity matrices 鈥� all achieved through meticulous matrix representation.

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Most popular questions from this chapter

Find a basis for the plane \(x-2 y+3 z=0\) in \(\mathbf{R}^{3}\). Then find a basis for the intersection of that plane with the \(x y\)-plane. Then find a basis for all vectors perpendicular to the plane.

Suppose \(A\) has a right-inverse \(B\). Then \(A B=I\) leads to \(A^{T} A B=A^{\mathrm{T}}\) or \(B=\left(A^{T} A\right)^{-1} A^{T}\), But that satisfies \(B A=I\); it is a left-inverse. Which step is not justified?

If \(w_{1}, w_{2}, w_{3}\) are independent vectors, show that the differences \(v_{1}=w_{2}-w_{3}\) \(v_{2}=w_{1}-w_{3}\), and \(v_{3}=w_{1}-w_{2}\) are dependent. Find a combination of the \(v\) 's that gives zero.

Find a basis for each of these subspaces of 3 by 3 matrices: (a) All diagonal matrices. (b) All symmetric matrices \(\left(A^{\mathrm{T}}=A\right)\). (c) All skew-symmetric matrices \(\left(A^{T}=-A\right)\).

Suppose \(T(M)=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right][M]\left[\begin{array}{cc}0 & 0 \\ 0 & 1\end{array}\right]\). Find a matrix with \(T(M) \neq 0\). Describe all matrices with \(T(M)=0\) (the kernel of \(T\) ) and all output matrices \(T(M)\) (the range of \(T\) ).
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