Linear Algebra -- Linear Transformations

[Julie]

I was hoping someone might have an idea on how to solve this problem. Any ideas? Thanks in advance, Julie.

Let T: Rⁿ --> R ^m be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation T(x)=0 (x is some vector, 0 is the zero vector) has a nontrivial solution.

A hint was given: Suppose u (a vector) and v (a vector) in Rⁿ are linearly independent and yet T(u) and T(v) are linearly dependent. Then
cT(u) + dT(v)=0 for some weights c and d, not both equal to zero. Use this equation.

 

[mark07]

Following the hint...

You have cT(u) + dT(v)=0 for some weights c and d, not both equal to zero.

Here is the magic of linearity:

0 = cT(u) + dT(v) = T(cu+dv)

x=cu+dv solves T(x)=0, and it's non-trivial because {u,v} is linearly independent (so non-zero vectors) and at least one of c or d is non-zero. Makes sense?

 

[Julie]

Yes it does, thank you very much