Question

If T1(x) and T2(x) are onto linear transformations from Rn to
Rm, then so is W(x) = T1(x) + T2(x).

True or False? can you explain why?

Answer 1

Test for closure under addition and scalar multiplication.

Answer 2

So T1(x) + T2(x) = T(x1+x2)?

and cT1(x) +cT2(x) = c(T1(x)+T2(x))?

Answer 3

If we know they are linear transformatoins, what's the point of checking for that?

Answer 4

or do I just check only for W(x)?

Answer 5

No, you test W for closure.
You check if it is true that W(x+y)=W(x)+W(y) and W(cx)=cW(x) by using the fact that T1 and T2 are linear transformations.

Answer 6

But there is no W(x+y) for this problem. It's only W(x). So what do I add W(x) to?

Answer 7

If it's confusing, then you can just use different letters and instead prove that W(u+v)=W(u)+W(v) and W(cu)=cW(u), where u and v are vectors in R^n.

Answer 8

OH, so it doesn't matter what vectors you put in there? You can have W(y) instead of W(x)?

Answer 9

Yeah.

Answer 10

So say I did W(x+y), does that mean I would have T1(x)+T2(x) + T1(y)+T2(y)?