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(Steps Shown) Let V be the vector space of all functions from the closed interval [-π, π] to R, and let W be the subspace spanned by the functions cos

Question: Let $V$ be the vector space of all functions from the closed interval $[-\pi, \pi]$ to $\mathbb{R}$, and let $W$ be the subspace spanned by the functions $\cos x, \sin x$, and the constant function 1 . Consider the function $T: W \rightarrow \mathbb{R}$ defined by

\[T(f)=\int_{-\pi}^{\pi} f(x) d x\]

Check that $T$ a linear transformation. What is the dimension of $W$ ? What is the dimension of the kernel of $T$ ? Find a basis for the kernel of $T$.

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(Steps Shown) Let V be the vector space of all functions from the closed interval [-π, π] to R, and let W be the subspace spanned by the functions cos

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