Linear operator L acts on vectors  , by mapping each such vector into the vector  


      L :    ---->  .


Therefore, the kernel of the operator L are those vectors   ,  for which  2x1 + x2 = 0.


They are all the vectors of the form  .


They form 1D-space (= actually, a line)  spanned on the vector (-1,2)  on the standard coordinate plane.

It is the same as to say that the kernel is the line  2y = -x  on standard (x,y) coordinate plane, or, equivalently,

the line  x + 2y = 0.


The image of this operator L is the 1D-space  {(x,0)}  - same as the x-axis on the standard coordinate plane (x,y).