- #1
iamalexalright
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Homework Statement
Let [tex]t \in L(V,W)[/tex]. Prove that t is an isomorphism iff it carries a basis for V to a basis for W.
Homework Equations
L(V,W) is the set of all linear transformations from V to W
The Attempt at a Solution
So I figured I would assume I have a transformation from a basis for V to a basis for W and prove that it is bijective (and hence isomorphic).
So let [tex]B_{v} = \{v_{i} | i \in I\}[/tex] be a basis for V and
[tex]B_{w} = \{w_{j} | j \in J\}[/tex] be a basis for W
So t is surjective if im(t) = W so...
[tex]im(t) = \{tv_{i} | v_{i} \in B_{v}\} =[/tex]
[tex]\{w_{j} | w_{j} \in B_{w}\}[/tex]
Since B_w spans W can I simply say then that, given above, the next line would be:
= W
so im(t) = W
Now to prove it is injective I need to show that ker(t) = {0}
[tex]tv_{i} = tv_{j}[/tex] implies
[tex]t(v_{i} - v_{j}) = 0[/tex] implies
[tex]v_{i} - v_{j} \in ker(t)[/tex]
If ker(t) = {0} then implies v_i = v_j but this doesn't seem like a formal way to prove this... any help?