The dimension of the span of the functions 1, cos(2x), and cos²(x) in the interval C[-π, π] is determined by their linear independence. The function cos(2x) can be expressed as a linear combination of the other two functions using the trigonometric identity cos(2x) = 2cos²(x) - 1. This indicates that the three functions are not linearly independent, confirming that the dimension of their span is 2. The Wronskian method is not necessary for this analysis, as a simpler approach using trigonometric identities suffices.
PREREQUISITESMathematicians, students studying linear algebra, and anyone interested in the properties of trigonometric functions and their applications in function spaces.
I realize that you already figured this out, but if you look at what I said, I didn't say that one of the functions was a multiple of another. I said that one function was a linear combination of the other two.Dustinsfl said: