Answered: Determine whether th e set ẞ is a basis…

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Determine whether th e set ẞ is a basis for the vector space V V = P 2,ẞ= {l, 1 + 2x + 3x2}

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 24CM

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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

Determine whether th e set ẞ is a basis for the vector space V V = P 2,ẞ= {l, 1 + 2x + 3x²}

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

ẞ comprises of the vectors 1 and 1 + 2x + 3x²

Step 2

Let's say a x 1 + b(1 + 2x + 3x²) = 0

Hence, (a + b) + 2bx + 3bx² = 0

Hence, a + b = 0

2b = 0

3b = 0

The only solution is the trivial solution that is a = b = 0

Hence the two vectors are linearly independent

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