There seems to be some inconsistencies in my mind that I'm trying to clear up, regarding the null space and the dimension theorem:
This is the problem:
Find a matrix whose null space is spanned by the vectors: $(2, −3, 1, 1, −1), (1, 0, −2, 1, 1), (2, −2, 1, 0, −1), (−8, 3, 1, 1, 1)$ in $\Bbb R^5$
Now, since the null space is four-dimensional, it leads me to believe that the rank of the sought matrix should be $1$, since I'm in $\Bbb R^5$. Basically, the matrix will correspond to a transformation whose image is one-dimensional.
The answer, which is given without any explanation, gives me a 2x5-matrix with a rank of $2$. After calculating the null space of that matrix, I find (just as I expected), a three-dimensional null space. What gives? Is this an error in my literature? Or am I missing some big chunk of theory? Or did I misunderstand the question? How would you go about finding this matrix?