Let T Rn rightarrow Rm be a linear transformation Given a s

Let T : R^n rightarrow R^m be a linear transformation. Given a subspace Z of R^m, let 5 {v elementof R^n | T(v) elementof Z). Show that S is a subspace of R^n.

Solution

Let v,w in S

Then T(v) and T(w) belongs to Z

Consider av+bw , a,b are scalars

We need to see if av+bw are in S

For this consider T(av+bw) = aT(v)+bT(w) (because T is linear transformtaion)

Now T(v) and T(w) are in Z

And so aT(v) and bT(w) are in Z

This => aT(v)+bT(w) are in Z

Hence we get that T(av+bw) is in Z

So we get av+bw is in S

Hence S is subspace of Rⁿ