Inverse mapping theorem , Transformations

[kingwinner]

A quick question this time...

Example: Let (u,v)=f(x,y)=(x-2y, 2x-y).
Find the region in the xy-plane that is mapped to the triangle with vertices (0,0),(-1,2),(2,1) in the uv-plane.


Solution:
(0,0)=f(0,0), (-1,2) = f(5/3,4/3), and (2,1)=f(0,-1), the region is the triangle with these vertices.


My question is:
Yes, we get three points, but how do you know FOR SURE that the region is a TRIANGLE? I am lost here...

Thanks for explaining!

 

[HallsofIvy]

You know "for sure" because the transformation is LINEAR- it maps straight lines into straight lines. If that's not enough, see what the transformation does to the line between (0,0) and (-1, 2) in the uv-plane. That is, of course, v= -2u.

Since u= x-2y and v= 2x-y, that becomes 2x-y= -2(x- 2y)= -2x+ 4y. Then adding y and 2x to both sides we have 4x= 5y or y= (4/5)x, the equation of a straight line.

Do the same with the other two sides of the triangle, v= (1/2)u and v= -(1/3)u+ 5/3 to see that they are mapped into straight lines.

 

[kingwinner]

Is there any quick way to see that this transfromation is linear?



"it maps straight lines into straight lines" <---is this always true for linear transfomations and is it what a linear transformation means geometrically? (I was never aware of the geometrical meaning of a linear transformation, I was just given the definition in my linear algebra course)

 

[HallsofIvy]

How can you be dealing with a problem like this and not know what a "linear" function is?
f(x,y)=(x-2y, 2x-y) is linear because it involves only sums and differences of products of numbers with a variable. There are no powers of variables, products of different variables, or more complicated functions. Notice that you could also write this as a single matrix multiplication:
$$f(x,y)= \left[\begin{array}{cc} 1 & -2 \\ 2 & -1\end{array}\right]\left[\begin{array}{c} x \\ y\end{array}\right]$$
That's a sure sign of a linear transformation.

Yes, it is fairly easy to prove that a linear transformation maps lines into lines.

 

[kingwinner]

But the definition of linear transformation that I've learned is:

T: U->V is a linear transformation iff T(au+bv)=aT(u)+bT(v) for all u E U, v E V, for all a, b E R

 

[Office_Shredder]

and a matrix multiplying vectors in Rⁿ satisfies that perfectly well

 

[kingwinner]

and a matrix multiplying vectors in Rⁿ satisfies that perfectly well

but by seeing that x-2y, 2x-y are linear polynominals, is it enough to say that f(x,y)=(x-2y, 2x-y) is a linear transformation? (these 2 definitions of "linear" seem quite distinct to me...for example, a linear polynominal allows a constant term, but a linear transfomration does not...)

 

[HallsofIvy]

Yes, but seeing that x-2y, 2x-y are linear polynomials without constant term is enough!