/*! 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 46 Prove Theorem 4.8: Suppose \(A\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 46

Prove Theorem 4.8: Suppose \(A\) and \(B\) are row canonical matrices. Then \(A\) and \(B\) have the same row space if and only if they have the same nonzero rows.

Short Answer

Expert verified
To prove Theorem 4.8, we showed that for row canonical matrices A and B: (1) if they have the same nonzero rows, they have the same row space, and (2) if they have the same row space, they have the same nonzero rows. Since their row echelon forms are unique, they must have the same nonzero rows when their row spaces are the same. Hence, A and B have the same row space if and only if they have the same nonzero rows.

Step by step solution

01

Part 1: Showing that if A and B have the same nonzero rows, then they have the same row space.

To show that if two row canonical matrices A and B have the same nonzero rows, then they have the same row space, we will follow these steps: 1. Since A and B are row canonical matrices, their nonzero rows form the bases for their respective row spaces. Recall that the row space is the subspace of linear combinations of the rows of the matrix. 2. Assume that A and B have the same nonzero rows. This means that any nonzero row of A is also a nonzero row of B, and vice versa. 3. Consequently, their bases consist of the same set of vectors, which means that their row spaces are the same. Therefore, if A and B have the same nonzero rows, then they have the same row space.
02

Part 2: Showing that if A and B have the same row space, then they have the same nonzero rows

To show that if two row canonical matrices A and B have the same row space, then they have the same nonzero rows, we will follow these steps: 1. Since A and B are row canonical matrices, their nonzero rows form the bases for their respective row spaces. 2. Assume that A and B have the same row space. This means that their row spaces have the same bases. 3. Since row canonical matrices have unique row echelon forms, A and B must have the same row echelon form, which means they have the same nonzero rows. Therefore, if A and B have the same row space, then they have the same nonzero rows. As we have shown both directions of the theorem, we can conclude that row canonical matrices A and B have the same row space if and only if they have the same nonzero rows.

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

Find a basis and the dimension of the subspace \(W\) of \(V=\mathbf{M}_{2,2}\) spanned by \\[A=\left[\begin{array}{rr} 1 & -5 \\ -4 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 1 \\ -1 & 5 \end{array}\right], \quad C=\left[\begin{array}{rr} 2 & -4 \\ -5 & 7 \end{array}\right], \quad D=\left[\begin{array}{rr} 1 & -7 \\ -5 & 1 \end{array}\right]\\]

Let \(V\) be the set of ordered pairs \((a, b)\) of real numbers. Show that \(V\) is not a vector space over \(\mathbf{R}\) with addition and scalar multiplication defined by (i) \((a, b)+(c, d)=(a+d, b+c)\) and \(k(a, b)=(k a, k b)\), (ii) \((a, b)+(c, d)=(a+c, b+d)\) and \(k(a, b)=(a, b)\), (iii) \((a, b)+(c, d)=(0,0)\) and \(k(a, b)=(k a, k b)\), (iv) \(\quad(a, b)+(c, d)=(a c, b d)\) and \(k(a, b)=(k a, k b)\).

Suppose \(W_{1}, W_{2}, \ldots, W_{r}\) are subspaces of a vector space \(V\). Show that (a) \(\operatorname{span}\left(W_{1}, W_{2}, \ldots, W_{r}\right)=W_{1}+W_{2}+\cdots+W_{r}\). (b) If \(S_{i}\) spans \(W_{i}\) for \(i=1, \ldots, r,\) then \(S_{1} \cup S_{2} \cup \ldots \cup S_{r}\) spans \(W_{1}+W_{2}+\cdots+W_{r}\).

In the space \(\mathbf{M}=\mathbf{M}_{2,3},\) determine whether or not the following matrices are linearly dependent: \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 0 & 5\end{array}\right], \quad B=\left[\begin{array}{rrr}2 & 4 & 7 \\ 10 & 1 & 13\end{array}\right], \quad C=\left[\begin{array}{rrr}1 & 2 & 5 \\ 8 & 2 & 11\end{array}\right]\) If the matrices are linearly dependent, find the dimension and a basis of the subspace \(W\) of \(\mathbf{M}\) spanned by the matrices.

Show that \(\operatorname{span}(S)=\operatorname{span}(S \cup\\{0\\}) .\) That is, by joining or deleting the zero vector from a set, we do not change the space spanned by the set.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.