Kindly solve the problems in the attached photo. Thank you! Let V and W be vecto

Question

Kindly solve

the problems in the attached photo. Thank you!

Let V and W be vector spaces, and let S : V rightarrow W and T : V rightarrow W be linear functions. Show that the set E = {x V | S(x) = T(x ) } is a sub vector space of V (i.e., show that if x, y E). then x + y E and that if c R then cx E). Let T : R3 rightarrow R2 be the linear function given by T([x y z]) = [1 2 3 4 5 6] [x y z] Determine the solution set K = {x | T(x) = ()}. Determine the solution set to T(x) = [6 15]. Let S : Rn rightarrow Rm and T : Rn rightarrow Rm be linear functions. Show that if S(e1) = T(e1) S(e2) = T(e2) ... S(en) = T(en) then S(x) = T(x ) for all x in Rn.

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