Linear Transformation I am new to linear algebra so I apologize beforehand for those of you who are math wizards. I need to know and understand why or why not for the following question:
T or F; $T (x,y) = (2x+5y,-x+2)$ is a linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$.
 A: Wizardry is not really required here.  Rather, what you need to know is the definition of a linear transformation: there are two properties to check.  
Do you know what these properties are?  If not, go back and take a look in your text.  (Or see Shaun Alt's answer.)
If you know what the properties are, did you try to check them?  If so, where did you get stuck?
A: We can use the definition of linear transformation (between vectors spaces like $\mathbb{R}^n$, though this can be generalized...):
$T$ is a linear transformation if for all vectors $\mathbf{u}$ and $\mathbf{v}$, $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$, and for all vectors $\mathbf{u}$ and real numbers $k$, we have $T(k\mathbf{u}) = kT(\mathbf{u})$.
Try to see if your $T$ satisfies this.  Note, since the domain is $\mathbf{R}^2$, you should take $\mathbf{u} = (u_1, u_2)$, and similar for $\mathbf{v}$.
A: Compute
$$
T((x,y) + (x',y')) = T(x+x', y+y')
$$
and compare with
$$
T(x,y) + T(x',y') \ ,
$$
for any real numbers $x,y,x',y'$.
Then do the same with
$$
T(\lambda (x,y)) = T(\lambda x , \lambda y) 
$$
and 
$$
\lambda T(x,y) \ ,
$$
for $\lambda $ any real number.
A: To show that something is a linear transformation, you have to work in generality to show that it satisfies the conditions in the definition of linear.  That is, you would have to check that for all $(x_1,y_1)$ and $(x_2,y_2)$ in $\mathbb R^2$ and for all $\lambda\in R$, $$T((x_1,y_1)+(x_2,y_2))=T(x_1,y_1)+T(x_2,y_2)\text{, and }$$
$$T(\lambda(x_1,x_2))=\lambda T(x_1,x_2).$$
On the other hand, to show that $T$ is not a linear transformation, you only need to show that there is one instance of a failure of a condition in the definition of linear, called a counterexample.  It is a good idea to try to make counterexamples as easy as possible, and with a numerical problem like this, that means you might want to try using lots of zeros, or maybe some ones.
