What it means to say that u and v are linearly DEPENDENT is

What it means to say that u and v are linearly DEPENDENT is that there is a number beta notequalto 0 so that u = beta v (u depends on v). Let T be a linear transformation. Show that if u and v are linearly dependent, then T(u) and T(y) are also linearly dependent.

Solution

u,v are linearly independent so there exists b so that

u=bv

T(u)=T(bv)=bT(v)

Hence, T(u) and T(v) are linearly dependent