/*! 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 22 Let \(B\) be an \(n \times m\) m... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 22

Let \(B\) be an \(n \times m\) matrix with rank \(m\). Prove that there exists an $m \times n\( matrix \)A\( such that \)A B=I_{m}$.

Short Answer

Expert verified
We are given a matrix \(B\) of size \(n \times m\) with rank \(m\), meaning that its columns are linearly independent. To prove that there exists a matrix \(A\) of size \(m \times n\) such that their product \(AB = I_m\), we consider the linear combination of columns of \(B\) with an unknown matrix \(A\). We find unique vectors \(a_1, a_2, \ldots, a_n\) such that their linear combinations with columns of \(B\) form the standard basis vectors in an \(m\)-dimensional space. By constructing the matrix \(A=[a_1 \ a_2 \ \cdots \ a_n]\), we show that there exists such an \(m\times n\) matrix \(A\) that satisfies the condition \(AB=I_m\).

Step by step solution

01

Recall definition of rank and properties of matrix multiplication

The rank of a matrix \(B\) is defined as the maximum number of linearly independent columns or rows it has. In this case, the rank of \(B\) is \(m\). Since the rank of \(B\) is equal to the number of its columns, all columns in \(B\) are linearly independent. Matrix multiplication has the following main properties: 1. Associativity: \((AB)C = A(BC)\) for compatible matrices \(A\), \(B\), and \(C\). 2. Distributivity: \(A(B+C) = AB+AC\) and \((A+B)C = AC+BC\) for compatible matrices \(A\), \(B\), and \(C\). 3. Compatibility with scalar multiplication: \(k(AB) = (kA)B = A(kB)\) for compatible matrices \(A\), \(B\) and scalar \(k\). 4. It is NOT commutative in general: \(AB \neq BA\) in general.
02

Reason about invertibility of \(B\)

Since the rank of \(B\) is \(m\), this means that the column space of \(B\) spans an \(m\)-dimensional space. In other words, the columns of \(B\) form a basis for an \(m\)-dimensional space. Now, let's consider the product \(AB\) with an unknown \(m\times n\) matrix \(A\). For \(AB\) to be equal to the identity matrix \(I_m\), we need to find such a matrix \(A\) that each of its columns, when multiplied by \(B\), will form the corresponding standard basis vector in an \(m\)-dimensional space.
03

Construct the matrix \(A\)

Let \(A\) be an \(m \times n\) matrix with columns denoted as \(a_i\), where \(i = 1,2, ..., n\). For \(AB\) to be equal to \(I_m\), we need to find such columns \(a_i\) that their linear combinations with columns of \(B\) form the standard basis vectors: $$Ba_1 = e_1, \ Ba_2 = e_2, \ ..., \ Ba_n = e_n$$ Since the columns of \(B\) form a basis for an \(m\)-dimensional space, there exists such a unique set of vectors \(a_1, \ a_2, \ ..., \ a_n\) that the above system of linear equations holds. So we can construct the matrix \(A\) from these vectors: $$A = [a_1 \ a_2 \ \cdots \ a_n]$$
04

Conclude the proof

We have constructed the matrix \(A\) such that \(A B=I_{m}\) by finding the set of vectors \(a_1, \ a_2, \ ..., \ a_n\), which form a linear combination with columns of \(B\) to obtain the standard basis vectors in an \(m\)-dimensional space. As these vectors exist due to the full rank of \(B\), we can conclude that there exists an \(m\times n\) matrix \(A\) such that \(AB=I_m\).

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

Determine whether the following vectors are solutions of \(x_{1}+2 x_{2}-4 x_{3}+3 x_{4}=15\) (a) \(u=(3,2,1,4)\) and (b) \(v=(1,2,4,5)\)

Consider the system $$\begin{array}{c}x+a y=4 \\\a x+9 y=b\end{array}$$ (a) For which values of \(a\) does the system have a unique solution? (b) Find those pairs of values \((a, b)\) for which the system has more than one solution.

In the notation of the open model of Leontief, suppose that $$ A=\left(\begin{array}{cc} \frac{1}{2} & \frac{1}{5} \\ \frac{1}{3} & \frac{1}{5} \end{array}\right) \quad \text { and } \quad d=\left(\begin{array}{l} 2 \\ 5 \end{array}\right) $$ are the input-output matrix and the demand vector, respectively. How much of each commodity must be produced to satisfy this demand?

Prove that for any \(m \times n\) matrix \(A, \operatorname{rank}(A)=0\) if and only if \(A\) is the zero matrix.

Describe the pivoting row-reduction algorithm. Also describe the advantages, if any, of using this pivoting algorithm.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

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.