kernel
The kernel of a transformation is the collection of all inputs that get crushed down to zero, that is, every vector x with T(x) = 0. It answers the question: which directions does this map throw away?
For a matrix, the kernel is exactly its null space, the solutions of A*x = 0. The two words name the same idea from two angles: kernel is the transformation viewpoint, null space is the matrix viewpoint.
The size of the kernel is a health check on the map. If the only thing sent to zero is the zero vector itself, the kernel is trivial and the transformation is one-to-one: distinct inputs stay distinct. A bigger kernel means information is being lost, because many inputs collapse onto the same output.
Flattening onto the x-axis erases the height, so the whole y-axis is sent to zero.
The kernel always contains the zero vector, because T(0) = 0 for any linear map. The real question is whether it contains anything else.