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August 31, 2020 7:49 am 30 pts
ransromaton from v, to V2 anstormation Example: Show that the mapping f: V.(F)> V,(F) which is defined by f(a,, a, a) = (a, a) is a homomorphism from V, onto V,. Also find its kernel. Solution. Let a = (a,, a,, a,) and b = (b,, b, b) be any two elements of the space V,(F) and a, be F, then f(aa+ bp) = f[(a(a,, a,, a,) + b(b,, b b.)] = f[(aa,, aa,, aa)+ (bb,, bb, bb,)] = f(aa, + bb,, aa,+ bb, aa, + bb) (aa, + bb,, aa, + bb) a(a,, b,) + b{b,, b) affa,, a, a,) + bf(b,, b, b.) af(a)+ bf(B) fis a homomorphism from space V, to V, a(b,b,b,) b(b 5b) Onto: Let (a,, a,) e V then corresponding to (a,, a), there exist (a,, a, 0) e V, for which fa a, 0) (a,, a) there exist f-pre image of each element of V, in V, f is onto homomorphism. Kernel: Let Ker f = K, then K will be the set of all the those vectors of V. which zero vector (0, 0) of V, i.e. (a exist (a,, a. 0) e V, for which f(a, h map on the K= {(a,, a, a) e V,I f(a,, a,, a,) = (0, 0)) f(a, a, a) (0, 0)(a, a,) = (0, 0) a, 0 and a, 0o K {(0, 0, a)a, e F But
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