Check Example 1.6. Example 1.6 Consider the space {a cos + b sin | a, b R} of functions of

Check Example 1.6.
Example 1.6
Consider the space {a ∙ cos θ + b ∙ sin θ | a, b ∈ R} of functions of the real variable θ. This is a natural basis (cos θ, sin θ) = (1 ∙ cos θ + 0 ∙ sin θ, 0 ∙ cos θ + 1 ∙ sin θ). A more generic basis for this space is (cos θ - sin θ, 2 cos θ + 3 sin θ). Verification that these two are bases is Exercise 27.