Linear Transformation and Linear dependence - Proof

[1nonly]

Homework Statement

Let T:Rⁿ to R^m 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.

 

[betel]

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.