Linear Transformation notation

[The Subject]

I'm confused about the notation
$$T:R^n \implies R^m$$
specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?

 

[FactChecker]

That is very possible. It is easy to define linear transformations that go from Rⁿ into a subspace of R^m, where m can be smaller, equal, or greater than n:

m smaller than n: (x1, x2) => x1
m equals n: (x1, x2) => (-x1, x1+x2)
m greater than n: (x1, x2) => (x1, x2, x1+x2); another is (x1, x2) => (x1, 0, 0)

 

[Mark44]

I'm confused about the notation
$$T:R^n \implies R^m$$
specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?

That's not the way to think about it. T isn't transforming a single number, like n. It's a map between a space of n dimensions to another space of m dimensions. If n = 2 and m = 3, T is a map from vectors in the plane to vectors in space (three dimensions).

Ordinary functions, which you're probably more familiar with, are maps from $\mathbb{R}^1$ to $\mathbb{R}^1$. (I added the 1 exponents only for emphasis.) A function of two variables is a map from $\mathbb{R}^2$ to $\mathbb{R}^1$.

 

[The Subject]

Okay makes sense. We're transforming vectors.

T is a map from vectors in the plane to vectors in space (three dimensions).

I don't feel confident with my understanding of this statement. What does it mean from vectors in the plane

 

[Mark44]

Okay makes sense. We're transforming vectors.

I don't feel confident with my understanding of this statement. What does it mean from vectors in the plane

Two-dimensional vectors, like <2, -1>. A vector in space would be, for example, <3, 1, 2>.