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If {v1, v2, . . . , vn} spans a vector space V, prove that some subset of the v’s is a basis for V.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 44E: Prove that in a given vector space V, the additive inverse of a vector is unique.

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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

If {v₁, v₂, . . . , v_n} spans a vector space V, prove that some subset of the v’s is a basis for V.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

Expert Solution

Step 1

Given that $\{v_{1}, v_{2}, . . . ., v_{n}\}$ spans a vector space $V$ .

We have to prove that some subset of $v' s$ is a basis for $V$ .

Step 2

Basis of a vector space:

Let $V$ be a $n$ -dimensional vector space and the vectors $\{v_{1}, v_{2}, . . . ., v_{n}\}$ is a basis of $V$ , then the vectors are linearly independent.

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$Vector Space$

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