In the context of 2-dimensional Euclidean space, has the space been transformed by the matrix in SVD? I guess not and I need a double check.

Question

I am learning svd

here is an geometric interpretation of SVD.

SVD could be viewed as some kind of linear transformation of a space.

denote the left one space_1, the right space_2.

Unit disc has been transformed with M from space_1 to space_2.

apparently the 2 vectors have been transformed, and the question is

is the space before svd same to the one after svd?

Has the space where the original vector red (1,0) and yellow (0, 1) live also been transformed? I guess not and I need a double check.

Answer 1

The terminology of "transformation" seems to make you think about this in a confusing way. In the picture, $M$ is map $\mathbb R^2\to\mathbb R^2$, where $\mathbb R^2$ denotes the euclidean plane with elements $(x,y)$ for real numbers $x$ and $y$. The space $\mathbb R^2$ stays the same, it never changes. However $M$ is like a machine taking an input point $(x,y)$ and producing an output point $M(x,y)$. You might say "$M$ transforms $(x,y)$ into $M(x,y)$", but the point $(x,y)$ has its own existence, regardless of you calculating $M(x,y)$ or not.

Now you can also take the unit disk $D$ consisting of all points $(x,y)$ with $x^2+y^2\le 1$ and apply $M$ to every point in the disk, call the set of outputs $M(D)$. Then $M(D)$ is the set of all points in the blue ellipse you see in the right hand side of the picture. It is still a subset of the same $\mathbb R^2$ as the original disk $D$, the picture just draws two copies of $\mathbb R^2$ so you don't get a circle overlapping with an ellipse. The disk $D$ still exists and hasn't been transformed in the sense that $D$ itself somehow changed, but applying $M$ to all points of $D$ you get the ellipse $M(D)$ as a result.

Answer 2

You may confused the vectors and the space where the vectors live.

In your case, the red vector \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix} yellow vector \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} and the transformed ones, all of them live in a same space, $R^2$.

the transformation affects the vectors rather than the space where the vectors live.

imagine you smack a clay ball to a square, you transformed the 3d vectors of the clay without affecting the real 3d world we people live.