Linear algebra: transformations

[seang]

Hello, I'll be online until I get this one completely figured out, so baby steps are for the win here.

Let L1:U->V and L2:U->W be linear transformations, and let L = L2 * L1 be the mapping defined by:

L(u) = L2(L1(u))

for each u which lies in U. Show that L is a linear transformation mapping U into W.
So basically, should I first show that L1(u) is a valid linear transform?, and then show that L2, is, too?

 

[radou]

Hm, I believe it's enough to assume L1 and L2 are linear transformations, and then proove that L is one, too.

 

[quasar987]

Is L2 from U to W or from V to W?

 

[seang]

V to W! thank you for that!
What about this? (in progress (latex noobie))

$$L(u) = L_2 (L_1 (u) )}$$

$$L(u_1 + u_2) = L_2 (L_1 (u_1 + u_2) )}$$

$$L(\alpha u_1 + \beta u_2) = L_2 (L_1 (\alpha u_1 + \beta u_2) )}$$

$$L(\alpha u_1 + \beta u_2) = L_2 (\alpha L_1(u_1) + \beta L_1 (u_2) )}$$

$$L(\alpha u_1 + \beta u_2) = \alpha L_2 (L_1(u_1)) + \beta L_2 (L_1(u_2))$$

$$\alpha L(u_1)+ \beta L(u_2) = \alpha L_2 (L_1(u_1)) + \beta L_2 (L_1(u_2))$$
Is this on the right track? If so, should I break it up into two pieces, and show that L1(u) is surely a mapping into V, and then show that L2(v) is surely a mapping into W?

Or am I way off

 

[HallsofIvy]

That's exactly right. Now all you have to do is note that
$$L_2(L_1(u_1))= L(u_1)$$
and
$$L_2(L_1(u_2))= L(u_2)$$