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  3. 1. let v and w be vector spaces over r. a) show th

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

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Advance Math

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

a) Show that if T: V ? W and S : V ? W are both lineartransformations, 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 ? ? R,then the map ?R: V ? W is given by (?R)(v) = ?(R(v)) is also alinear transformation.

c) Let E(V) be the set of all linear operators T: V ? V. Checkthat E(V) is a vector space with the addition and scalarmultiplication defined above.

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

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