- #1

1nonly

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## Homework Statement

Let T:R

^{n}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

c

_{1}u

_{1}+ c

_{2}v

_{2}= 0 where c1,c2 = 0

T(c

_{1}u

_{1}+ c

_{2}v

_{2}) = T(0) where c1 or c2 /= 0

## The Attempt at a Solution

Since we know:

T(c

_{1}u

_{1}+ c

_{2}v

_{2}) = T(0) where c

_{1}or c

_{2}/= 0

T(c

_{1}u

_{1}) + T(c

_{2}v

_{2}) = T(0)

c

_{1}T(u

_{1}) + c

_{2}T(v

_{2}) = T(0)

c

_{1}T(u

_{1}) + c

_{2}T(v

_{2}) = 0 (c

_{1}or c

_{2}/= 0)

__T(x) = 0 has trivial solution__

c

_{1}T(x) = 0 where c

_{1}/= 0

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