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Basis

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Hi.

I have the following 2 questions.

1. Find a vector that, together with vectors [1,1,1], and [1,2,1], forms a basis of R^3.

2. Show that the functions (c1 + c2sin^2x + c3cos^2x) form a vector space. Find a basis for it. What is its dimension?

Thanks

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Solution Summary

This provides examples of finding a basis.

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1. you need only find a vector which,

together with (1, 1, 1) and (1, 2, 1)

forms a basis for R^3.

The GS process is not necessary here,

since there's no stipulation on the

orthogonality of these vectors.

Let's try (1, 0, 0); then we need only

show that the equation

a(1, 0, 0) + b(1, 1, 1) + c(1, 2, 1) = 0

implies a = b = c = 0

But the above vector equation can be

written

(a, 0, 0) + (b, b, b) + (c, 2c, c) = ...

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