Possible Images of a Linear Map T:R4->R4

[Aceuniverse]

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!

 

[Dick]

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!

The rank of a matrix is the number of linearly independent vectors that span the image space. I.e. the number of linearly independent columns of the matrix of T. I'm really not sure how to answer in a more clear way.

 

[Aceuniverse]

The rank of a matrix is the number of linearly independent vectors that span the solution space. I.e. the number of linearly independent columns of the matrix of T. I'm really not sure how to answer in a more clear way.

Oh it's not the definition of rank I am concerned with. I wanted to know if using rank was a good method to answering the question "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. "