Recall the linear transformation 𝑅𝜃 : ℝ2→ℝ2 that rotates every 𝑥⃗∈R2 by a fixed angle 𝜃

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Recall the linear transformation 𝑅𝜃 : ℝ2→ℝ2 that
rotates every 𝑥⃗∈R2 by a fixed
angle 𝜃 counter-clockwise about the origin. Now,
let 𝑣⃗∈ℝ3 be non-zero and fix 𝜃∈ℝ. We define the
linear operator 𝑅𝑣⃗,𝜃 : ℝ3→ℝ3 to be the counter-clockwise
rotation by 𝜃 about the line spanned by 𝑣⃗. That
is, 𝑅𝑣⃗,𝜃 takes any
vector 𝑥⃗ in ℝ3 and rotates it
counter-clockwise about the line spanned by 𝑣⃗.
(a) Determine the standard matrix of 𝑅𝑒⃗3,𝜃. Justify your
answer.
(b) Prove that 𝑅𝑣⃗,𝜃 has 11 as an eigenvalue
for every 0≠𝑣⃗∈ℝ3.