In one of 3Blue1Brown's videos on linear algebra, he states that when a space such as a 2d plane is mapped to a line, a line of vectors in the plane is squished to a point, such as in his example for the null space shown below.
However, I read in another
post that the null space is the set of all vectors which lose their identity when a transformation is applied to them. The user defined 'losing their idenity' as being set the zero vector, however I would have imagined that the other lines in the plane which are squished to other, non-zero points on the line would also be 'losing their identity'. So my questions are:
- Are the other lines squished to points other than zero considered to be in the null space? (I assume not)
- If not, what is the significance of knowing this particular line of vectors which become the zero vector?
I believe the dimension of the null space is the dimension lost if the linear transformation has less than full rank but don't know why it is important otherwise. I would have thought that knowing all the vectors which can not be accessed anymore would have been more important, rather than just the ones which now land on the zero vector. I hope you understand what I am trying to get at, and sorry if it isn't clear enough.
Best Answer
Regarding the two questions:
Answer: You are right, the answer is not.
Answer: For linear mappings, $v$ is in the null space of $A$, if $Av=0$. Therefore, for any other vector $u$, $A(u+v)=Au+Av=Au$. In other words, if you translate any vector $u$ along any vector in the null space, from the linear transformation perspective, nothing will happen. In other words, everything that lies on the null-space of translated by $u$, will have the same value under the linear transformation $x\to Ax$.