4. Suppose that V and W are vector spaces over a field F and let L:V → W be a function. a) Prove that L is a linear transformation if and only if L(Au + v) = AL(u)+ L(v) for all A E F and u, v E V. b) Let v1,..., Vn E V and A1,..., An E F. If L is a linear transformation, prove that L(A1v1 +...+ An Vn) = A1L(v1) +...+ An L(vn). (Hint: induction.)

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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4. Suppose that V and W are vector spaces over a field F and let L :V → W be a function.
a) Prove that L is a linear transformation if and only if
L(Au + v) = \L(u) + L(v)
for all A E F and u, v E V.
b) Let v1,..., Vn E V and A1,.….., An E F. If L is a linear transformation, prove that
L(A1v1 + ...+AnVn) = A1L(v1)+...+ AnL(Vn).
(Hint: induction.)
Transcribed Image Text:4. Suppose that V and W are vector spaces over a field F and let L :V → W be a function. a) Prove that L is a linear transformation if and only if L(Au + v) = \L(u) + L(v) for all A E F and u, v E V. b) Let v1,..., Vn E V and A1,.….., An E F. If L is a linear transformation, prove that L(A1v1 + ...+AnVn) = A1L(v1)+...+ AnL(Vn). (Hint: induction.)
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