Let V and W be vector spaces over a field F let alpha Epsilo

Let V and W be vector spaces over a field F. let alpha Epsilon Hom(V, W) and beta Epsilon Hom(W, V) satisfy the condition that alpha beta alpha = alpha. If w Epsilon im(alpha), show that alpha^-1 (w) = {beta(w) + upsilon - beta alpha(upsilon) Epsilon V}.

Solution

Let x 2 V , x = a1v1 + a2v2 + · · · + anvn for some scalars a1, a2, . . . , an.
UT(x) = UT(a1v1 + a2v2 + · · · + anvn)
= U(a1T(v1) + a2T(v2) + · · · + anT(vn)
= U(a1x1 + a2x2 + · · · + anxn)
= a1U(x1) + a2U(x2) + · · · + anU(xn)
= a1v1 + a2v2 + · · · + anvn
= a
Let x 2 V , x = b1x1 + b2x2 + · · · + bnxn for some scalars b1, b2, . . . , bn.
TU(x) = TU(b1x1 + b2x2 + · · · + bnxn)
= T(b1U(x1) + b2T(x2) + · · · + bnT(xn)
= T(b1v1 + b2v2 + · · · + bnvn)
= b1T(v1) + b2T(v2) + · · · + bnT(vn)
= b1x1 + b2x2 + · · · + bnxn
= a

we found that it is true .