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# linear transformations watch

1. i am stuck on this question and was wondering if i could get any help:

the equation of a circle is x^2 + y^2 = 1. a curve is defined as C' = f(C), which is the image of C under the linear transformation f represented by the matrix:

2 1
1 2

i.e. the inverse image of C' (the curve) is C (the circle).

Find the equation of the curve C'.

i have a linear transformation of f(x,y) = (2x + y, x + 2y) and i can also work out the inverse and determinant of the matrix. but i dont know how to put it all together to get the equation of the curve help is much appreciated
2. Hmm working from first principles here ..

If you had a point (x,y) on C', you could work out from which point on C it had appeared, by inverting your matrix.

If that point were (p,q), you would know that p^2 + q^2 = 1 for it to be on C.

Is that enough??
3. if you're allowed to use a parametrisation you can sub x for cost and y for sint and set 0<t<2pi
4. (Original post by ian.slater)
Hmm working from first principles here ..

If you had a point (x,y) on C', you could work out from which point on C it had appeared, by inverting your matrix.

If that point were (p,q), you would know that p^2 + q^2 = 1 for it to be on C.

Is that enough??
i am not following i cant do that because i dont have the equation of the curve. can you give a bit more help?
5. By inverting your matrix, you can find out the point that was transformed to say (x,y)

(p,q) = M^-1 (x,y)

So you have p and q in terms of x and y.

For (p,q) to be on C, p^2 + q^2 = 1. Which is now an equation connecting x and y.

6. (Original post by ian.slater)
By inverting your matrix, you can find out the point that was transformed to say (x,y)

(p,q) = M^-1 (x,y)

So you have p and q in terms of x and y.

For (p,q) to be on C, p^2 + q^2 = 1. Which is now an equation connecting x and y.

thanks for the help, i eventually got it. can you quickly check for me to see if the answer is 5x^2 + 5y^2 - 8xy - 9 = 0? thanks again
7. (Original post by wrooru)
can you quickly check for me to see if the answer is 5x^2 + 5y^2 - 8xy - 9 = 0?
Agreed.
8. (Original post by wrooru)
thanks for the help, i eventually got it. can you quickly check for me to see if the answer is 5x^2 + 5y^2 - 8xy - 9 = 0? thanks again
I get the same answer. I'm also wondering whether the same result can be obtained by thinking geometrically - i.e. what does the (2,1/1,2) matrix represent in terms of shear and stretch?
9. I get eigenvalues of 1 and 3. The eigenvector corresponding to 3 is (1,-1). So the transformation is a stretch of factor 3 along the axis y=-x

That leads to 9(x+y)^2 + (x-y)^2 = 18 which is the same equation. Seems like more work though.
10. ..
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