What Are the Possible Forms of the Kernel in Linear Transformations from R3?

[nehap.2491]

Let T:R3 to V be a linear transformation from R3 to any vector space.Show that the kernel of T is a line through the origin, a plane through the origin,the origin only, or all of R3.

 

[micromass]

The kernel of a linear map is always a subspace.

So you'll need to show that the subspaces of $$\mathbb{R}^3$$ are
- the origin
- lines through the origin
- planes through the origin
- the space itself.

 

[Outlined]

I would say, as T is a linear map, that it depends on what T does with e₁ = (1, 0, 0), e₂ =(0, 1, 0) and e₃ = (0, 0, 1).

It is possible that the kernel equals R³. For example the linear map which maps everything to 0 in V.