Note: the two proofs are not the same!

[franz32]

How will I prove that...
Show that L: V -> W is a linear transformation if and only if
L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any
vectors u and v in V.

For L(au +bv), this is my proof. (Is this wrong?)

L(au + bv) = L [ a(a', b', c') + b(a'', b'', c'')]
= L [ aa' + ba'', ab' + bb'', ac' + bc'' ]
= (aa' + ab' + ac') + ( ba" + bb" +bc")
= a(a' + b' +c') =b(a" + b" + c")
= aL(u) + bL(v)

 

[cookiemonster]

I've never studied what you're doing formally, but I've always thought that what you're "proving" is the definition of a linear transformation. Why would you have to prove an arbitrary definition?

cookiemonster

 

[franz32]

Here's...

Yeah that's right. It's an arbitrary definition of the linear transformation. My professor wants me to do it...

In the textbook I'm using, it looks like this

1. L(u+v) = L(u) + L(v)
2. L(ku) = kL(u)

 

[cookiemonster]

Well, since it's an arbitrary definition, I don't really see the point of "proving" it.

The only thing I can imagine him doing is asking you to combine the two conditions as it's usually stated. I usually see it in this form:

$$L[\boldsymbol{v_1}+\boldsymbol{v_2}] = L[\boldsymbol{v_1}]+L[\boldsymbol{v_2}]$$
$$L[a\boldsymbol{v}] = aL[\boldsymbol{v}]$$

My guess is that he wants to see you combine these.

Edit: Just noticed you typed the form yourself. Guess I should read a little more slowly next time...

cookiemonster

 

[turin]

I'm with cookiemonster. There is no such thing as proving a definition aside from showing the entry in a dictionary.

 

[Hurkyl]

Show that L: V -> W is a linear transformation if and only if
L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any
vectors u and v in V.

Well, as has been pointed out,

"L: V -> W is a linear transformation" means
1. L(u+v) = L(u) + L(v)
2. L(ku) = kL(u)
for all vectors u and v in V and scalars k.

The problem is asking you to prove

L: V -> W is a linear transformation

if and only if

L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any
vectors u and v in V.


So, you start with the assumption that "L: V -> W is a linear transformation" then prove "L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any vectors u and v in V."

Then, (as a separate piece of work!) you start with the assumption "L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any vectors u and v in V." and prove "L: V -> W is a linear transformation".