# Linear transformations

## Homework Statement

T:$${R^3 \rightarrow {R^2}$$ given by $$T(v_1,v_2,v_3) = (v_3 -v_1, v_3 - v_2)$$

If linear, specify the range of T and kernel T

The attempt at a solution
Okay, I went ahead and tried to find the kernel of T like here:
\begin{align*}&v_3 - v_1 = 0\\ &v_3 - v_2 = 0\end{align*}

Thus, \begin{align*}&v_3 = v_1 \\ &v_3 = v_2\end{align*}

So choosing v3 as s gives the 1-D basis of W= s(1, 1, 1) **a column vector**

But I'm not entirely sure how to get the range. IF I did the kernel correctly, then that means the dimension of the range will be 2 as 2+1 = 3 (the dimension of the domain). But when I try to do the range, I get a 3-dimensional basis where v1,v2,and v3 are their own LI vectors as so:
(y1,y2) = s(1,1) + t(-1,0) + r(0,-1)

Any help?

Related Calculus and Beyond Homework Help News on Phys.org
Everything seems fine. So where's your difficulty?

HallsofIvy
Homework Helper
You have correctly deduced that the range must have dimension 2 and you know that the range is a subspace of R2.

How many subspaces of dimension 2 do you think R2 has!