1. Let V and W be vector spaces over R. a) Show that if T: V...

1. Let V and W be vector spaces over R.

a) Show that if T: V → W and S : V → W are both linear transformations, then the map S + T : V → W given by (S + T)(v) = S(v) + T(v) is also a linear transformation.

b) Show that if R: V → W is a linear transformation and λ ∈ R, then the map λR: V → W is given by (λR)(v) = λ(R(v)) is also a linear transformation.

c) Let E(V) be the set of all linear operators T: V → V. Check that E(V) is a vector space with the addition and scalar multiplication defined above.

d) Suppose dim V = n. What is dim(E(V))? Justify your answer.

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Expert Solution

Colby Messinger answered 1 year ago

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