Which of the following are linear transformations from R2 to R2?(a) Shear: T(x, y) = (x+ cy, y) for some constant c.(b) Translation: T(x, y) = (x + a, y + b) for some constantsa and b.(c) Blow-up: T(r, y) = (ax, by) for some constants a and b.(d) Rotation: if r = r cos 0, y = rsin 0, thenT(z,y) = (r cos(60 + ), r sin(0 + )),

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Asked Oct 29, 2019
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Nedd help with d)

Which of the following are linear transformations from R2 to R2?
(a) Shear: T(x, y) = (x+ cy, y) for some constant c.
(b) Translation: T(x, y) = (x + a, y + b) for some constantsa and b.
(c) Blow-up: T(r, y) = (ax, by) for some constants a and b.
(d) Rotation: if r = r cos 0, y = rsin 0, then
T(z,y) = (r cos(60 + ), r sin(0 + )),
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Which of the following are linear transformations from R2 to R2? (a) Shear: T(x, y) = (x+ cy, y) for some constant c. (b) Translation: T(x, y) = (x + a, y + b) for some constantsa and b. (c) Blow-up: T(r, y) = (ax, by) for some constants a and b. (d) Rotation: if r = r cos 0, y = rsin 0, then T(z,y) = (r cos(60 + ), r sin(0 + )),

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Expert Answer

Step 1

(d) To determine if the given mapping T is a linear transformation on the plane R^2

Step 2

Recall the definition of a linear transformation. We will show that a rotation as defined in d) is indeed a linear transfromation by verifying 1) and 2)

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Definition: A mapping T: R->R2,is linear if 1)T(u+v) Tu+Tv.Vu,ve R 2)T(cu) xTu,ce R, Vu e R

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Step 3

d) write the mapping in terms of th...

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d)T(x, y) (r cos cos -r sin0sinø rsin0cos rcos@sin p (rcos-ysin,xsinp+ycos ) () (note that o,cos p,sino have fixed values)

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