Find the linear or affine transformations that satisty the desired properties and write it in the form T(x) = Ax + b: The transformation T : mathbb{R}

Ramsey 2020-11-02 Answered
Find the linear or affine transformations that satisty the desired properties and write it in the form T(x)=Ax + b:
The transformation T : R2  R2 sending the origin to itself and a triangle of vertices
(0, 0), (1, 0), (0, 1) to a triangle of verices (0, 0),
(22, 22), (22, 2).
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Expert Answer

Khribechy
Answered 2020-11-03 Author has 100 answers

We have the tranformation as,
T(t1 : t2)=(1)det(0xyzt1a1b1c1t2a2b2c21111)det(a1b1c1a2b2c2111)
Graphical representation:
image
The red triangle is transformed into blue triangle under the transformation T.
Here,
T(t1 : t2)=(1)det(0xyzt1a1b1c1t2a2b2c21111)det(a1b1c1a2b2c2111)
=det(0xyzt1010t20011111)det(010001111)
= xt1  yt1 + xt2  x  zt21
=(y  x)t1 + (z  x)t2 + x
Now, plugging in the transformed entries.
=((2222)  (00))t1 + ((2222)  (00))t2 + (00)
=(2222)t1 + (2222)t2

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