Let f = cos2x, g = sin2x. Which of the following lie in the space spanned by f and g
Question
Let f = cos2x, g = sin2x.
Which of the following lie in the space spanned by f and g?
Solution
The space spanned by f and g includes all linear combinations of f and g. In other words, any function that can be written as af + bg for some constants a and b lies in the space spanned by f and g.
For example, the function h = cos2x + sin2x lies in the space spanned by f and g because it can be written as 1f + 1g. Similarly, the function i = 3cos2x - 2sin2x also lies in the space spanned by f and g because it can be written as 3f - 2g.
On the other hand, a function like j = sin(x) does not lie in the space spanned by f and g because it cannot be written as a linear combination of cos2x and sin2x.
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