Transformations of vectors

Question

So i was watching some videos online about Transformations and vector valued functions. Now the conclusion that i came with was that a transformation (lets say (T: ${R^2}$ $\rightarrow$ ${R^2}$) ) is actually a transformation of 2 dimensional space, where if given a vector input, it shows the "movement of the vector" withing the transformation of the two dimensional space. Essentially a morphing of space is happening and as space bends, shrinks, turns etc we see where the vector lands and thats our transformation.

Now my question is this reasoning correct? , as other places seem to only state that a transformation is a function that takes a vector and spits out another vector.

Also if my reasoning is correct would a transformation across dimensions e.g (T: ${R^3}$ $\rightarrow$ ${R^2}$) ) be conceptualised as transforming three dimensional space into two dimensional space??.

Answer 1

Yes, your reasoning is correct.

The transformation deforms each individual input vector to produce an output vector.

But we can also think about what happens when the totality of all input vectors is deformed simultaneously. Then, the transformation does in fact deform space (in some sense).

As you say, a mapping from $\mathbb{R}^3$ to $\mathbb{R}^2$ actually "squashes" all the vectors of $\mathbb{R}^3$ onto a plane.

When thinking about transformations, it's often a good idea to figure out the effect of the deformation on some simple shape, like a square. A linear mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$ will always deform a square into a parallelogram. The parallelogram might be a degenerate one, like a line or even a single point, but still a parallelogram in some sense. Similarly, a linear mapping from $\mathbb{R}^3$ to $\mathbb{R}^3$ will always deform a cube into a parallelipiped.

For more general transformations, it helps to think of space as a cube of jello. The transformation will bend/warp/stretch/twist the cube. Any points or lines or vectors inside the jello will get carried along as the jello deforms.

There is an excellent series of YouTube videos that provide insight into the geometry of linear algebra.