Question

Linear Transformation Question

Answer 1

Describe all linear transformations from \[\mathbb{R}^{2}\] to \[\mathbb{R}\].
What do their graphs look like?

Answer 2

Also, how do I clean up this latex? As in, how do I make sure it doesn't make space?

Answer 3

http://www.math.utah.edu/~richins/teaching/2270/test1solns.pdf
4 is the solution, but why is A is the 1x2 matrix? I was under the impression because you're going from R^2 to R, from two dimensions to one dimension, the surface would become a line. (a plane to a line)

Answer 4

Let me ask you a side question :
How does the graph of a function from R^2 to R like : \(f(x,y) = x^2 + y^2\) look like ?

Answer 5

The latex advise... b/c igtg soon.

if you type `\(\mathbb{R}^{2}\)` then you get \(\mathbb{R}^{2}\)

if you type `\(\mathbb{R}\)` then you get \(\mathbb{R}\)

Answer 6

In general if you type something in equation editor, and then after entering the equation(s) entirely, you can change `\[ \]` to `\( \)` and that will allow you to write on the same line with latex.

Sometimes you will need to add a `\displaystyle` or such....

Answer 7

gtg gb

Answer 8

If I view \(f(x,y)=x^2 + y^2 \) (thanks @SolomonZelman !) in a two dimensional manner, I would see small planes, so in the same vein, the answer would obviously be that I would view planes!

Answer 9

Let me understand some theory of Calc III again. If f(x,y) is free to choose whatever value it wants, doesn't it really constitute as another variable? So that, z = x^2 + y^2 is identical to f(x,y) = x^2 + y^2?
Am I misunderstanding this? Thinking of it as another variable, I think of it being a three-D object going down to a two-D object, so the object would have to be a plane.

I don't understand how this could relate back to linear transformations.
(if you think this is completely wrong to the point it's unexplainable to me, tell me! I'll pursue help in our math lab when it opens!)

Answer 10

f(x,y) = x^2+y^2 represents a "surface" in 3D

Answer 11

Since the domain is 2D and image is 1D, the graph is a surface in 3D (the function maps each point in xy plane to a real number)