Linear Transformation and Linear dependence - Proof

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Homework Statement


Let T:Rn to Rm be a linear transformation that maps two linearly independents vectors {u,v} into a linearly dependent set {t(u),T(v)}. Show that the equation T(x)=0 has a nontrivial solution.

Homework Equations



c1u1 + c2v2 = 0 where c1,c2 = 0

T(c1u1 + c2v2) = T(0) where c1 or c2 /= 0

The Attempt at a Solution



Since we know:

T(c1u1 + c2v2) = T(0) where c1 or c2 /= 0
T(c1u1) + T(c2v2) = T(0)
c1T(u1) + c2T(v2) = T(0)
c1T(u1) + c2T(v2) = 0 (c1 or c2 /= 0)

T(x) = 0 has trivial solution
c1T(x) = 0 where c1 /= 0

I'm not sure how to connect those two ideas or if there even relevant to the solution proof.
 

Answers and Replies

  • #2
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I'm not sure what your last expression is supposed to mean, but the block of four equations is already the proof. Maybe it is easier to see it, if you read it backwards.
 

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