Let T R5 rightarrow R3 be a linear transformation whose null

Let T: R^5 rightarrow R^3 be a linear transformation whose null space is a three dimensional subspace of R^5. What is the dimension of the image of T? Describe the image geometrically Is the image of T a subset of R^5 or R^3?

Solution

Theorem:

If T::V->W is a linear transformation , then

dim V = dim Null space T + dim Image T

In the present case, dim V=5, dim Nullspace is 3 (given)

So dim (Image T) (using the theorem ) is 2.

Geometrically , Image of T is a two dimensional subspace (a plane) of R³

Image of T is a subspace (hence a subset of R³)