A student learning about linear independence makes the following inaccurate statement: "A subset S of a vector space V is linearly independent if, for any list of distinct vectors u1, u2,..., Un E S, the equation c1u1 + c2u2 + · · · + CnUn = 0 has the trivial solution." The student's statement is incorrect because the word "only" has been left out at a critical place. Explain, using a few sentences, why the student's statement is incorrect as it is written. Please be specific in your explanation; simply saying that a word is missing from the definition is not a sufficient explanation.

A student learning about linear independence makes the following inaccurate statement: "A subset S of a vector space V is linearly independent if, for any list of distinct vectors u1, u2,..., Un E S, the equation c1u1 + c2u2 + · · · + CnUn = 0 has the trivial solution." The student's statement is incorrect because the word "only" has been left out at a critical place. Explain, using a few sentences, why the student's statement is incorrect as it is written. Please be specific in your explanation; simply saying that a word is missing from the definition is not a sufficient explanation.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

A student learning about linear independence makes the following inaccurate statement:
"A subset S of a vector space V is linearly independent if, for any list of distinct
vectors u1, u2,..., Un E S, the equation c1u1 + c2u2 + · · · + CnUn = 0 has the trivial solution."
The student's statement is incorrect because the word "only" has been left out at a critical place. Explain,
using a few sentences, why the student's statement is incorrect as it is written. Please be specific in your
explanation; simply saying that a word is missing from the definition is not a sufficient explanation.

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