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 B 0 so that u Bv (u depends on v). Let T be a linear transformation. Show that if u and v are linearly dependent, then T(u) and T(v) are also linearly dependent

Solution

If u and v are linearly dependent means there exist some b so that

u=bv

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

ie T(u)=bT(v)

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