ToT. (o and 8 are called conjugate elements) Let o,TE Sn. Define 6 = Show that if o (i) = j, then 6 = (t(i)) = T(j). (a) (b) Explain that the previous part says that if you apply T to each of the entries in the cycle notation of o, then you get the cycle notation for ô. In other words, if o has cycle decomposition (а, а .. ак, )(b bz ... bk,).... Then 8 has cycle decomposition (143 8) (2 6 5), (163) 7 5 2), Illustrate the previous part by letting (c) σ Ξ and quickly writing down the cycle decomposition for toT1. Check your

Use mathematical induction to prove that if L is a linear transformation from V to W, then L (α1v1 + α2v2 +· · ·+αnvn) = α1L (v1) + α2L (v2)+· · ·+αnL (vn)

2. Suppose T : Rª → Rª is a linear transformation, and suppose we know -2] 3 10 T -3 and T 7 Then T 1 3 -8 a) is equal to 7 -13 b) is equal to -8- -69 3 10 c) is equal to -8

Suppose that T: R" → Rm is defined by T() = A for each of the following matrices below: [40] E = 0 0 02 0 D = 0 -1 0 0 0 0.5 [201] F = 10 3 01 (a) Rewrite T: R" → Rm with correct numbers for m and n for each transformation. What is the domain and codomain of each transformation? (b) Find some way to explain in words and/or graphically what each transformation does as it takes vectors from R to Rm. You might find it t helpful to try out a few input vectors and see what their image is under the transformation. This might be difficult, but an honest effort will give you credit. (c) For the transformation, can you get any output vector? (Any vector in R™) i. If so, explain why you can get any vector in Rm. ii. If not, give an example of an output vector you can't get with the transformation and explain why.