Explanation Given information: a matrix U is in row echelon form. Proof: A basis for a vector space V is a set of vectors that (1) is linearly independent and (2) spans V . We shall show that the nonzero row vectors of the row echelon matrix U satisfies these two properties for the row space of U. Linear independence: If there are no nonzero rows in U , then U is the zero matrix and hence its row space is {0}. The set of nonzero row vectors in this case is then the empty set. This is trivially a basis for {0} (by definition). Now suppose that U has at least one nonzero row. Because U is in row echelon form, if it has k nonzero rows, then those rows will be the first k rows of U . the leading entry of the first row of U has zero everywhere beneath it in the matrix U . hence it cannot be formed by a linear combination of the other rows. Thus the first row is linearly independent of the others. Now we can remove the first row and apply the above argument to the new first row. Applying an induction argument, we can see that all of nonzero rows are linearly independent from the others. Hence, the set of nonzero row vectors is linearly independent...

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

9th Edition

ISBN: 9780321962218

Author: Steven J. Leon

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 3.6, Problem 36E

To determine

To show: if a matrix U is in row echelon form, then the nonzero row vectors of U form a basis for the row space of U .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.6, Problem 35E

Chapter 3, Problem 1E

Chapter 3 Solutions

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Ch. 3.1 - Consider the vectors x1=(8,6)T and x2=(4,1)T in R2...Ch. 3.1 - Repeat Exercise 1 for the vectors x1=(2,1)T and...Ch. 3.1 - LetC be the set of complex numbers. Defineaddition...Ch. 3.1 - Show that mn together with the usual additionand...Ch. 3.1 - Show that C[a,b] , together with the usual...Ch. 3.1 - LetP be the set of all polynomials. Show that P,...Ch. 3.1 - Show that the element 0 in a vector space...Ch. 3.1 - Let x, y, and z be vectors in a vector space V....Ch. 3.1 - Let V be a vector space and let xV . Show that (a)...Ch. 3.1 - Lei S be the set of all ordered pairs of real...

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

if a matrix U is in row echelon form, then the nonzero row vectors of U form a basis for the row space of U .

Solution Summary: The author demonstrates that the nonzero row vectors of the row echelon matrix U satisfies these two properties for the space of U.

Videos

To show: if a matrix U is in row echelon form, then the nonzero row vectors of U form a basis for the row space of U .

Want to see the full answer?

Chapter 3 Solutions