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Let V be the set of real numbers regarded as a vector space over the field of rational

Chapter 1, Problem 3

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QUESTION:

Let V be the set of real numbers regarded as a vector space over the field of rational numbers. Prove that V is infinite-dimensional. Hint: 62 Chap. 1 Vector Spaces Use the fact that IT is transcendental, that is, it is not a zero of any polynomial with rational coefficients.

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QUESTION:

Let V be the set of real numbers regarded as a vector space over the field of rational numbers. Prove that V is infinite-dimensional. Hint: 62 Chap. 1 Vector Spaces Use the fact that IT is transcendental, that is, it is not a zero of any polynomial with rational coefficients.

ANSWER:

Step 1 of 2

We have

Take   then

is neither represented as

Where    and   nor represented as a linear combination of rational.

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