Assume that T is a linear transformation. Find the standard matrix of T. T:R2⇒R2 first reflects points through the vertical x2−aξs and then reflects points through the line X2=X1. A=?

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Assume that T is a linear transformation. Find the standard matrix of T.   T:R2⇒R2 first reflects points through the vertical x2−aξs and then reflects points through the line X2=X1.   A=?

Jonathan Burroughs · Accepted Answer

Step 1  It is given that T is linear transformation  T:R2⇒R2  The standard matrix of  T which firstly reflects the points through vertical x2−aξs 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 x2−aξs.  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=[01−10]

Fasaniu · Accepted Answer

Step 1We are looking for a&amp;nbsp;2×2&amp;nbsp;matrix that reflects points through the horizontal axis and then reflects points through the line&amp;nbsp;x2=x1.First the points need to be reflected through the horizontal&amp;nbsp;x1−aξs. This then means that the&amp;nbsp;x1−coordinate&amp;nbsp;remains unaffected and the&amp;nbsp;x2−coordinate&amp;nbsp;changes sign:R1=[100−1]Note: if you multiply this matrix by the vector&amp;nbsp;(x1,x2)T&amp;nbsp;to the right, then you obtain the vector&amp;nbsp;(x1,−x2)T.Step 2Thus, the points need to be reflected through the line&amp;nbsp;x2=x1. The&amp;nbsp;x1−coordinate&amp;nbsp;and&amp;nbsp;x2−coordinate&amp;nbsp;then interchange.R2=[0110]Note: if you multiply this matrix by the vector&amp;nbsp;(x1,x2)T&amp;nbsp;to the right, then you obtain the vector&amp;nbsp;(x2,x1)T.Step 3The result of the reflection matrices, with the first reflection matrix on the right, is then the combination of the reflections.T=R2×R1=[0110]×[100−1]=[0−110]Note: if you multiply this matrix by the vector&amp;nbsp;(x1,x2)T&amp;nbsp;to the right then you obtain the vector&amp;nbsp;(−x2,x1)T

Assume that T is a linear transformation. Find the standard

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