Why is a certain matrix transformation is $\mathbb{R}^2$ to $\mathbb{R}^2$?

Question

Given a matrix transformation $\begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}$, why does it go from $\mathbb{R}^2$ to $\mathbb{R}^2$, when it squishes the plane into a line?

There is nothing in the word "transformation" that implies "onto". — Gerry Myerson, Jun 25 '17 at 03:39
The line ${(0,y) : y\in \mathbb R}$ is a subset of $\mathbb R^2$ so this map does map $\mathbb R^2$ into $\mathbb R^2$. — User8128, Jun 25 '17 at 03:41

score 1 · Accepted Answer · answered Jun 25 '17 at 03:41

1

The image of the point $(a,b) $ is $$f (a,b)=(0a+1.b,0a+0b)=(b,0) $$

all images are in the line whose equation is $y=0$.

$f $ is from $\mathbb R^2$ to $\mathbb R^2$ but $f (\mathbb R^2)=\{(x,0) : x\in \mathbb R\} $ is a line.

$f $ is not surjective.

answered Jun 25 '17 at 03:41

hamam_Abdallah

62,951

Why is a certain matrix transformation is $\mathbb{R}^2$ to $\mathbb{R}^2$?

1 Answers1