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## Homework Statement

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.

## Homework Equations

N/a

## 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!