Linear Transform: Proving Action Determined by Basis

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SUMMARY

The discussion centers on proving that a linear transformation T: U → V is fully determined by its action on a basis B = {u₁, u₂, ..., uₙ} of the domain U. The initial approach highlights that since a basis spans U and is linearly independent, the transformation T can be defined entirely by its effects on the basis vectors. A more rigorous proof is suggested, starting with the assumption that T(B) is known, allowing for the expression of T(u) for any vector u in U.

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Homework Statement



Prove the following:

The action of a linear transformation [tex]T:U\rightarrow V[/tex] is completely determined by its action on a basis [tex]B=\left\{u_1,u_2,\text{...},u_n\right\}[/tex] for the domain U.

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The Attempt at a Solution



Okay, I feel like my solution is too simple. It doesn't feel rigorous enough, but I'm not sure how to improve it. Any ideas?

By definition, the linear transformation [tex]T:U\rightarrow V[/tex] only acts on the domain U. Also by definition, a basis of U must also span U and be linearly independent. Thus, U is completely determined by B, and in turn, T is completely determined by its action on B. QED
 
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You haven't really proven it yet.
I'd start by: suppose that T(B) is given, i.e. [itex]T(u_i) = v_i[/itex]. Let [itex]u \in U[/itex]. You should now be able to write down T(u).
 

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