null space
The null space of a matrix A is the collection of all input vectors x that A sends to zero, that is, every x solving A*x = 0. It is exactly the kernel of A viewed as a transformation: the inputs the matrix crushes flat to the origin.
It is always a subspace: the zero vector is in it, and any combination of null-space vectors is still sent to zero. Its dimension has a name, the nullity, and it counts how many independent directions get destroyed.
The null space tells you the freedom hiding in solutions of A*x = b. If the null space is just the single point zero, any solution is unique. But if the null space contains a whole line, you can slide any solution along that line and still solve the equation, so there are infinitely many answers.
Every multiple of (1,-1) is sent to zero, so the null space is that whole line.
If a matrix has any non-zero null space, it cannot be inverted, because more than one input maps to the same output.