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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0Can you find even one such transformation? – 2012-05-09
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0That's what i am asking. i think answer should be zero. But how to show? – 2012-05-09
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3Remember 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. – 2012-05-09
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1If you have some ideas about the problem (like that the answer is zero), then *put them in your question*. – 2012-05-09
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0i have added sir..it should be either 0 , 1 , 3 or 3! – 2012-05-09
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0Can you work out how many linear transformations there are that satisfy just the first two of your three equations? – 2012-05-09
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0what should be approach? – 2012-05-09
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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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2I think a hint would probably have been more helpful than a complete answer. (I'm just bothering to say this because I upvoted your answer, and then realised that I don't like the fact that complete answers to easy questions get disproportionately many upvotes, so upvoting it perhaps didn't make sense.) – 2012-05-09
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0@Tara B: Unless the OP merely copies the above answer, there is work for the OP to do in order to see that the above *does* answer the question. – 2012-05-09
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0@TaraB Actually, I agree with you, I probably should have just given a hint. Sometimes, my eagerness to answer overrides the filter that asks, "What kind of answer is best for this user?" I'll leave it as is, since I'll guess the OP has already read it, but will keep trying to improve my instincts for future answers. – 2012-05-09
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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