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Chapter 4: Q29E (page 200)

State true or false, if a, b andare distinct real numbers then the polynomials a basis of linear space V then(x-b)(x-c),(x-a)(x-c) and(x-a)(x-b) must be a linear independent.

Short Answer

Expert verified

The statement is True.

Step by step solution

01

Determine the property of independent.

Consider the real number a,b,c such that a , b and c are distinct real numbers.

The set of polynomial are linear independent if the root corresponding to each polynomial are distinct.

02

Determine the statement is True or False

As a, b and c are distinct real numbers implies (x - a), (x - b) and (x - c) are distinct real numbers.

If (x - a), (x - b) and (x - c) are distinct real numbers then (x - b) (x - c), (x - a)(x - c) and (x - a)(x - b) are distinct polynomials.

The roots corresponding to each polynomial is distinct.

By the definition of the linear independent, the set of polynomials (x - b) (x - c), (x - a) (x - c) and (x - a) (x - b) are linear independent.

Hence, the statement is True.

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