/*! 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} Q33E Let \({M_{2 \times 2}}\) be the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Q33E (page 191)

Let \({M_{2 \times 2}}\) be the vector space of all \(2 \times 2\) matrices, and define \(T:{M_{2 \times 2}} \to {M_{2 \times 2}}\) by \(T\left( A \right) = A + {A^T}\), where \(A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\).

  1. Show that \(T\)is a linear transformation.
  2. Let \(B\) be any element of \({M_{2 \times 2}}\) such that \({B^T} = B\). Find an \(A\) in \({M_{2 \times 2}}\) such that \(T\left( A \right) = B\).
  3. Show that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\) with the property that \({B^T} = B\).
  4. Describe the kernel of \(T\).

Short Answer

Expert verified
  1. It is proved that \(T\) is a linear transformation.
  2. \(T\left( A \right) = B\)
  3. It is proved that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\) with the property \({B^T} = B\).
  4. The kernel of \(T\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}0&b\\{ - b}&0\end{array}} \right):b\,\,{\mathop{\rm real}\nolimits} } \right\}\).

Step by step solution

01

Show that \(T\) is a linear transformation

a)

Let \(A\) and \(B\) be any matrix in \({M_{2 \times 2}}\). Then,

\(\begin{array}{c}T\left( {A + B} \right) = \left( {A + B} \right) + {\left( {A + B} \right)^T}\\ = A + B + {A^T} + {B^T}\\ = \left( {A + {A^T}} \right) + \left( {B + {B^T}} \right)\\ = T\left( A \right) + T\left( B \right).\end{array}\)

Let \(c\) be any scalar. Then,

\(\begin{array}{c}T\left( {cA} \right) = {\left( {cA} \right)^T}\\ = c{\left( A \right)^T}\\ = cT\left( A \right).\end{array}\)

Therefore, \(T\) is a linear transformation.

Thus, it is proved that \(T\) is a linear transformation.

02

Determine that \(A\) in \({M_{2 \times 2}}\) such that \(T\left( A \right) = B\)

b)

Consider \(B\) is an element of \({M_{2 \times 2}}\) with \({B^T} = B\), and \(A = \frac{1}{2}B\). Then,

\(\begin{array}{c}T\left( A \right) = A + {A^T}\\ = \frac{1}{2}B + {\left( {\frac{1}{2}B} \right)^T}\\ = \frac{1}{2}B + \frac{1}{2}{B^T}\\ = \frac{1}{2}B + \frac{1}{2}B\\ = B.\end{array}\)

Thus, \(T\left( A \right) = B\).

03

Show that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\) with the property \({B^T} = B\)

c)

Part b demonstrates that the range of \(T\) contains the set of all \(B\) in \({M_{2 \times 2}}\) with \({B^T} = B\).

It must be demonstrated that \(B\) in the range of \(T\) has this property.

Suppose \(B\) is in the range of \(T\), then \(B = T\left( A \right)\) for some \(A\) in \({M_{2 \times 2}}\). Therefore,\(B = A + {A^T}\) and

\(\begin{array}{c}{B^T} = {\left( {A + {A^T}} \right)^T}\\ = {A^T} + {\left( {{A^T}} \right)^T}\\ = {A^T} + A\\ = A + {A^T}\\ = B.\end{array}\)

Thus, \(B\) has the property \({B^T} = B\).

Hence, it is proved that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\), with the property \({B^T} = B\).

04

Describe the kernel of T

d)

Consider \(A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\) is in the kernel of T. Then \(T\left( A \right) = A + {A^T} = 0\), and

\(\begin{array}{c}A + {A^T} = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right) + \left( {\begin{array}{*{20}{c}}a&c\\b&d\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{2a}&{c + b}\\{b + c}&{2d}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}0&0\\0&0\end{array}} \right).\end{array}\)

By solving, you get \({\mathop{\rm a}\nolimits} = d = 0\) and \({\mathop{\rm c}\nolimits} = - {\mathop{\rm b}\nolimits} \).

Therefore, the kernel of \(T\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}0&b\\{ - b}&0\end{array}} \right):b\,\,{\mathop{\rm real}\nolimits} } \right\}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(calculus required) Define \(T:C\left( {0,1} \right) \to C\left( {0,1} \right)\) as follows: For f in \(C\left( {0,1} \right)\), let \(T\left( t \right)\) be the antiderivative \({\mathop{\rm F}\nolimits} \) of \({\mathop{\rm f}\nolimits} \) such that \({\mathop{\rm F}\nolimits} \left( 0 \right) = 0\). Show that \(T\) is a linear transformation, and describe the kernel of \(T\). (See the notation in Exercise 20 of Section 4.1.)

Suppose \({{\bf{p}}_{\bf{1}}}\), \({{\bf{p}}_{\bf{2}}}\), \({{\bf{p}}_{\bf{3}}}\), and \({{\bf{p}}_{\bf{4}}}\) are specific polynomials that span a two-dimensional subspace H of \({P_{\bf{5}}}\). Describe how one can find a basis for H by examining the four polynomials and making almost no computations.

Question 11: Let\(S\)be a finite minimal spanning set of a vector space\(V\). That is,\(S\)has the property that if a vector is removed from\(S\), then the new set will no longer span\(V\). Prove that\(S\)must be a basis for\(V\).

If a\({\bf{6}} \times {\bf{3}}\)matrix A has a rank 3, find dim Nul A, dim Row A, and rank\({A^T}\).

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

15. Let \(A\) be an \(m \times n\) matrix, and let \(B\) be a \(n \times p\) matrix such that \(AB = 0\). Show that \({\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B \le n\). (Hint: One of the four subspaces \({\mathop{\rm Nul}\nolimits} A\), \({\mathop{\rm Col}\nolimits} A,\,{\mathop{\rm Nul}\nolimits} B\), and \({\mathop{\rm Col}\nolimits} B\) is contained in one of the other three subspaces.)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.