(I think) My textbook says something is a linear transformation if
- $L(ax) = aL(x)$
- $L(x+y) = L(x) + L(y)$
- $L(x) = A(x)$
But a lot of sites I've been on haven't proved these 3 things, so I just wanted to make sure that this is the proper way to prove it.
For example, if $L(x) = (x_1, x_2, x_1 + 2x_2)^T$, then is it a linear transformation from $R^2$ to $R^3$? I know that it is, but I'm not sure if what I have above is how I'm supposed to prove it.
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$\begingroup$This is almost right, but that third condition shouldn't be there (in fact, I don't know what it means in general--what is "$A$"?). That is, a map $L$ between two vector spaces is a linear transformation if and only if it satisfies $L(ax)=aL(x)$ and $L(x+y)=L(x)+L(y)$ (for any scalar $a$ and any elements $x,y$ in the domain of $L$).
You can check this directly for your example $L(x)=(x_1,x_2,x_1+2x_2)$ (for $x=(x_1,x_2)$). For any scalar $a$, we have $ax=(ax_1,ax_2)$, so $$L(ax)=L(ax_1,ax_2)=(ax_1,ax_2,ax_1+2ax_2)=a(x_1,x_2,x_1+2x_2)=aL(x).$$
If $x=(x_1,x_2)$ and $y=(y_1,y_2)$, then $x+y=(x_1+y_1,x_2+y_2)$, so $$\begin{align} L(x+y)=L(x_1+y_1,x_2+y_2)={} &(x_1+y_1,x_2+y_2,x_1+y_1+2x_2+2y_2)\\={}&(x_1,x_2,x_1+2x_2)+(y_1,y_2,y_1+2y_2)\\={}&L(x)+L(y). \end{align}$$
As you can see, checking something like this can be a kind of lengthy computation sometimes! But it's not hard--you just plug in the definitions and rearrange the terms appropriately.
$\endgroup$ $\begingroup$Yes your textbook is right, basically a function is a linear transformation if and only if scalar multiplicity is reserved meaning that letting $a $ be a real number then
$L(a*x)=a*L(x)$
In your example if you wanted to show this property holds you show that
$2L(x)=2(x_1,x_2,x_1+2x_2)=(2x_1,2x_2,2x_1+4x_2)$
The second property that linear transformations must satisfy is preservation or distribution over vector addition. Let's say $v$ and $u$ are vectors then
$L(x+v)=L(x)+L(v)$
Meaning you can add the vectors and then transform them or you can transform them individually and the sum should be the same. If in any case it isn't, then it isn't a linear transformation.
The third property you mentioned basically says that linear transformation are the same as matrix transformations. So every linear transformation has a matrix corresponding to it called the standard matrix. Hope this helps.
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