Assume that T is a linear transformation. Find the standard

Marenonigt 2021-12-17 Answered
Assume that T is a linear transformation. Find the standard matrix of T.
T:R2R2 first reflects points through the vertical x2aξs and then reflects points through the line X2=X1.
A=?
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Expert Answer

Jonathan Burroughs
Answered 2021-12-18 Author has 37 answers
Step 1
It is given that T is linear transformation
T:R2R2
The standard matrix of T which firstly reflects the points through vertical x2aξs and then reflects points through the line x2=x1.
Let us consider the standard basis of R2 that is
{e1=(1,0),e2=(0,1)}
Step 2
Reflection of point along vertical x2aξs.
For any
(x1,x2)R2
T(x1,x2)=(x1,x2)
Firstly finding the images of standard basis under this transformation.
T(1,0)=(1,0)
T(0,1)=(0,1)
Step 3
Reflection points through the line x2=x1.
For any (x1,x2)R2
Y(x1,x2)=(x2,x1)
Now finding the images of obtained points under the reflection of points along x2=x1.
T(1,0)=(0,1)
T(0,1)=(1,0)
Therefore, the matrix of required linear transformation is given by
A=[0110]

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Fasaniu
Answered 2021-12-19 Author has 46 answers

Step 1
We are looking for a 2×2 matrix that reflects points through the horizontal axis and then reflects points through the line x2=x1.
First the points need to be reflected through the horizontal x1aξs. This then means that the x1coordinate remains unaffected and the x2coordinate changes sign:
R1=[1001]
Note: if you multiply this matrix by the vector (x1,x2)T to the right, then you obtain the vector (x1,x2)T.
Step 2
Next the points need to be reflected through the line x2=x1. The x1coordinate and x2coordinate then interchange.
R2=[0110]
Note: if you multiply this matrix by the vector (x1,x2)T to the right, then you obtain the vector (x2,x1)T.
Step 3
The combination of the reflections is then the product of the reflection matrices, with the first reflection matrix right.
T=R2×R1=[0110]×[1001]=[0110]
Note: if you multiply this matrix by the vector (x1,x2)T to the right then you obtain the vector (x2,x1)T

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