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

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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.
 
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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.
 
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.
 

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