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Please advise why I am wrong.

Thanks

Asif

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__Problem statement:__Let T: U->V be an isomorphism. Let U1, U2,...,Un be linearly independent. Show that T(U1), T(U2),...,T(Un) is linearly independent in V.

__Problem solution__1- In U, this is true: (lambda[1])(U[1]) + ... (lambda[n])(U[n]) = 0 as this is linearly independent and all lambdas are 0 (for linear independence)

2- Since it is an isomorphism, every vector in U uniquely maps to V.

3- Therefore V is linearly independent also.

4- v1 = (alpha[1])T(U[1])

vn = (alpha[n])T(U[n])

Since v is lineraly independent:

0 = alpha[1]T(U[1]) + alpha[2]T(U2)+....+alpha[n]T(U[n])

My questionMy question

a- Is step 3 a valid assumption

b- By saying equation in step 4, can I safely assume that T(u1),...,T(un) are linearly independent?

Thanks

Asif