Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Q38E (page 191)

[M] Repeat Exercise 37 for three random-integer-valued \(5 \times 7\) matrices \(A\) whose ranks are 5, 4, and 3. Make a conjecture about how \(CR\) is related to \(A\) for any matrix \(A\). Prove your conjecture.

Short Answer

Expert verified

The weights of \({r_j}\) are needed to construct the \({j^{th}}\) column of \(A\) from the columns of \(C\) and \(C{{\mathop{\rm r}\nolimits} _j} = {{\mathop{\rm a}\nolimits} _j}\).

Step by step solution

01

Write the matrix as in Exercise 37

Consider matrix\(A\)as\(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

Matrix \(C\) is \(C = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&5&{ - 3}\\{ - 4}&6&{ - 2}&{ - 5}\\5&{ - 7}&5&2\\{ - 3}&5&{ - 1}&{ - 4}\\6&{ - 8}&4&9\end{array}} \right]\).

Matrix \(R\) is \(R = \left[ {\begin{array}{*{20}{c}}1&0&{\frac{{13}}{2}}&0&5&0&{ - 3}\\0&1&{\frac{{11}}{2}}&0&{\frac{1}{2}}&0&2\\0&0&0&1&{\frac{{ - 11}}{2}}&0&7\\0&0&0&0&0&1&1\end{array}} \right]\).

02

Make a conjecture about how \(CR\) is related to \(A\)

When \(A\) is nonzero, \(A = CR\). \(CR = \left[ {\begin{array}{*{20}{c}}{C{{\mathop{\rm r}\nolimits} _1}}&{C{{\mathop{\rm r}\nolimits} _2}}& \cdots &{C{{\mathop{\rm r}\nolimits} _n}}\end{array}} \right]\), where \({{\mathop{\rm r}\nolimits} _1},...,{{\mathop{\rm r}\nolimits} _n}\) represent the columns of \(R\). The columns of \(R\) may or may not have the pivot columns of \(R\). Let the first column of \(R\), \({e_i}\) be the \({i^{th}}\) pivot column of \(R\); the \({i^{th}}\) column in the identity matrix. Thus, \(C{e_i}\) is the \({i^{th}}\) pivot column of \(A\). Multiply \(C\) with the pivot column of \(R\) and the result is the corresponding pivot column of \(A\) in its pivot position because both \(A\) and \(R\) have the pivot columns in the same location.

Consider \({r_j}\) as a non-pivot column of \(R\). So, \({r_j}\) contains the weights needed to construct the \({j^{th}}\) column of \(A\) from the pivot columns of \(A\) as explained in examples 9 and 10 of section 4.3. Therefore, the weights of \({r_j}\) are used to construct the \({j^{th}}\) column of \(A\) from the columns of \(C\) and \(C{{\mathop{\rm r}\nolimits} _j} = {{\mathop{\rm a}\nolimits} _j}\).

Thus, \({r_j}\) contains the weights needed to construct the \({j^{th}}\) column of \(A\) from the columns of \(C\) and \(C{{\mathop{\rm r}\nolimits} _j} = {{\mathop{\rm a}\nolimits} _j}\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

In Exercise 3, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

3. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{4}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{2}}\\{ - {\bf{2}}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{0}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{0}}\\{ - {\bf{1}}}\end{array}} \right)\)

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 \(T\).

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

16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

(Hint: Write \(A + B\) as the product of two partitioned matrices.)

If A is a \({\bf{4}} \times {\bf{3}}\) matrix, what is the largest possible dimension of the row space of A? If Ais a \({\bf{3}} \times {\bf{4}}\) matrix, what is the largest possible dimension of the row space of A? Explain.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.

Sign-up for free