1. The problem statement, all variables and given/known data 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. 2. Relevant equations N/a 3. 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!