# Counting number of linear transformations

Let $v_{1} = (1, 0)$, $v_{2} = (1, -1)$ and $v_{3} = (0, 1)$. How many linear transformations $T :\mathbb {R^2}\rightarrow \mathbb {R^2}$ are there such that $T(v_{1} ) = v_{2}$, $T(v_{2} ) = v_{3}$, $T(v_{3} ) = v_{1}$. I am finding difficulty in tackling to this problem. I tried to identify corresponding linear transformation. But didn't come to any conclusion.It should be either 0, 1, 3 or $3!$

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Can you find even one such transformation? – Chris Eagle May 9 '12 at 12:46
That's what i am asking. i think answer should be zero. But how to show? – srijan May 9 '12 at 12:48
Remember that once you have defined a linear map on a basis, you can work out what its value has to be on any other vector. – Matthew Pressland May 9 '12 at 12:49
If you have some ideas about the problem (like that the answer is zero), then put them in your question. – Chris Eagle May 9 '12 at 12:49
i have added sir..it should be either 0 , 1 , 3 or 3! – srijan May 9 '12 at 12:51

$v_2 = v_1-v_3$, so you'd need: $$v_3 = T(v_2) = T(v_1-v_3)=T(v_1)-T(v_3) = v_2-v_1$$ which is not true.