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 $3!$
Counting number of linear transformations
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linear-algebra
matrices
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0Do you know that multiplication by a $2\times2$ matrix is a linear transformation, and that every linear transformation is multiplication by some matrix? I'm trying to find out what you actually know about linear algebra, so I know where to pitch an answer (and you are not being very forthcoming). – 2012-05-09
1 Answers
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$v_2 = v_1-v_3$, 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.
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1@AndréNicolas: That's true, but I think there's still more benefit to be gained from working out the rest from a hint than from figuring out why an answer works. – 2012-05-09