Question

In plain english, how would you define linear transformations?

Answer 1

That depends on how deep you've gotten in regards to linear transformations.

Answer 2

If you look at my question, which is currently above yours, it is also on linear transformations, but notice how mine has degrees of 2. I might confuse you there if you don't know how you can have a linear transformation using polynomials of higher powers.

Answer 3

I've done problems in the Linear Algebra book by David Lay. I get confused by T: R^n --> R^m. Doesn't n usually refer to the number of columns in the matrix and m = the number of rows?

Answer 4

sry, wrong question

Answer 5

ah, ok, you need to forget the part about focusing on m X n

Yes m is the number of rows and n is the number of columns usually

Answer 6

R^n-->R^m represents moving from one space to another. Kind of like y=f(x) where you have a number x. You "Transform" it using the f operator and go into the y "space".

Answer 7

You will still need to create matricies but will need to be more open mminded on the n-->m since sometimes to write the vectors in columns and sometimes in rows. There is also the transpose of a matrix that switches the columns of a matrix with its rows and vice-versa

Answer 8

I'm just wondering if I'm thinking correctly about how linear transformations work. So, basically, you start off with a set of vectors (or lines) in some defined dimension, could be 2d, 3D or whatever. The goal is to transform those vectors into some other set of vectors (in some dimension), by multiplying the original vector by a matrix (I'll call A).

Answer 9

That is one, I guess you could call it type, of transformation. There is also differentiation, integration, and so on. You still move from one space to another, but it is by means of changing one set to another set. You don't always deal with matricies.

Answer 10

oh, ok Thanks.

Now when I see the notation x --> Ax. Is that the pretty much the same as T: R^n --> R^m?

Answer 11

In the earlier parts of linear transformations yes. Later on, you have polynomial vector spaces and more abstract things

Answer 12

OK thanks. I appreciate it.