(Solution) - Check Example 1 6 Example 1 6 Consider the space a cos -(2025 Original AI-Free Solution)

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