So here is a theoretical question. Let L1 and L2 be linear transformation from a vector space V into Vector space W. Let {v1, v2, ..., vn} be a basis for V. Show that if L1 (vi) = L2 (vi) for i=1,2, ..., n then L1 (v) = L2 (v) for any v in V ...?

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Mathematics

Mira Osborn

16 November, 11:32

So here is a theoretical question. Let L1 and L2 be linear transformation from a vector space V into Vector space W. Let {v1, v2, ..., vn} be a basis for V. Show that if L1 (vi) = L2 (vi) for i=1,2, ..., n then L1 (v) = L2 (v) for any v in V ...?

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Dixie · Answer 1

if v belongs to V, then we can find scalars a1, a2, ..., an, such that v=a1*v1+a2*v2 + ... + an*vn, L1 (v) = L1 (a1*v1+a2*v2 + ... + an*vn) = a1*L1 (v1) + a2*L1 (v2) + ... + an*L1 (vn) = a1*L2 (v1) + a2*L2 (v2) + ... + an*L2 (vn) = L2 (a1*v1+a2*v2 + ... + an*vn) = L2 (v)