Show that a linear map T:R4->R4 has one of the following as its image: just the origin 0, a line through 0, a plane through 0, a copy of R3 through 0, or all of R4.
The Attempt at a Solution
I'm not sure I'm even understanding the problem, I asked a friend (honestly) and he said that the Rank of T when it is a matrix can determine whether it maps to just to the origin or all of R4. it went something like this:
Rank : 0 -> Maps to origin
Rank: 1 -> Maps to a line through 0
Rank:2 -> Maps to a plane through 0
Rank:3 -> Maps to a copy of R3
Rank:4 -> Maps to all of R4
my only concern is how does this answer the question, and if this is wrong how should I approach it? Any help would be appreciated, thanks in advance!