Let @ be a fixed angle. If x is in R2, let L(x) be the element of R2 obtained by

Question

Let @ be a fixed angle. If x is in R2, let L(x) be the element of R2 obtained by rotating x through the angle @. Assume L: R2 --> R2 is linear. Let u1 = e1, u2 = e2. Then u1, u2 form a basis for R2. Find a formula for L(x) using the following theorem (hint: first determine v1 = L(u1), v2 = L(u2)).

Thm: Let U, V be vector spaces over F. If U is finite-dimensional, u1,...,un is a basis for for U and v1,...,vn are any n vectors in V, not necessarily linearly independent, then there exists a unique linear mapping L: U --> V such that L(uk) = vk for k in {1,...,n}.

Explanation / Answer

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