Prove a set of vectors is linearly independent

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To prove that the set {L(v1),...,L(vn)} is linearly independent given that {v1,...,vn} is linearly independent and L is a one-to-one linear mapping, it must be shown that the equation a1 L(v1) + ... + an L(vn) = 0 implies a1 = a2 = ... = an = 0. Since L is one-to-one, if L(v1),...,L(vn) were linearly dependent, it would contradict the linear independence of {v1,...,vn}. Therefore, the linear independence of the original set ensures the linear independence of the mapped set. A rigorous proof must demonstrate this relationship clearly using the definition of linear independence. The conclusion is that the linear independence of the original set is preserved under one-to-one linear mappings.
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Homework Statement


Suppose that {v1,...,vn} is a linearly independent set in a vector space V and let L:V --> W be a one-to-one linear mapping. Prove that {L(v1),...,L(vn)} is linearly independent.


Homework Equations


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The Attempt at a Solution


If L is a one-to-one linear mapping, then it maps a specific vector in V to a specific vector in W, therefore, {L(v1),...,L(vn)} must be a linearly independent set of vectors.
However, this is not the rigorous proof that the question is looking for.
I will appreciate any help possible, thank you~~~
 
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Hint: a set {u1, ..., un} of vectors is linearly independent iff the equation
a1 u1 + ... + an un = 0
can only be satisfied by a1 = a2 = ... = an = 0.
 

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