# Proving linear transformation

1. Feb 24, 2004

### 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)

2. Feb 24, 2004

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?

3. Feb 24, 2004

### 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)

4. Feb 24, 2004

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...

5. Feb 25, 2004

### turin

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

6. Feb 25, 2004

### Hurkyl

Staff Emeritus
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 seperate 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".