Section 30 number 21 space V over a field F with a vector space V' over the same field F.Theory21. Próve that if V is a finite-dimensional vector space over a field F, then a subset {B, B2, Bn) of V is a basisfor V over F if and only if every vector in V can be expressed uniquely as a linear combination of the B,.

Name: Section 30 number 21 space V over a field F with a vector space V' ove
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Section 30 number 21 space V over a field F with a vector space V' over the same field F.Theory21. Próve that if V is a finite-dimensional vector space over a field F, then a subset {B, B2, Bn) of V is a basisfor V over F if and only if every vector

Answered: Prove that ifV is a finite-dimensional…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 44EQ

See similar textbooks

Related questions

Concept explainers

Topic Video

Question

Section 30 number 21

space V over a field F with a vector space V' over the same field F.
Theory
21. Próve that if V is a finite-dimensional vector space over a field F, then a subset {B, B2, Bn) of V is a basis
for V over F if and only if every vector in V can be expressed uniquely as a linear combination of the B,.

Expert Solution

Step 1

First, show that if each vector in $V$ can be uniquely expressed as a linear combination of the $β_{i}$ ,then $\{β_{1}, β_{2}, . . ., β_{n}\}$ is a basis for $V$ .

Suppose that each vector in $V$ can be uniquely expressed as a linear combination of the $β_{i}$ .

Then the set of vectors $\{β_{1}, β_{2}, . . ., β_{n}\}$ generates V .

Since $0 = 0 \cdot β_{1} + 0 \cdot β_{2}, . . . + 0 \cdot β_{n}$ , this expression is the unique linear combination of the $β_{i}$ that yields the zero vector.

Hence, the vector $β_{i}$ , are linearly independent.

Trending now

This is a popular solution!

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Discrete Probability Distributions$

Learn more about

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

Similar questions