Marenonigt

2021-12-17

Assume that T is a linear transformation. Find the standard matrix of T.

$T:{R}^{2}\Rightarrow {R}^{2}$ first reflects points through the vertical ${x}_{2}-a\xi s$ and then reflects points through the line $X}_{2}={X}_{1$ .

$A=?$

Jonathan Burroughs

Beginner2021-12-18Added 37 answers

Step 1

It is given that T is linear transformation

$T:{R}^{2}\Rightarrow {R}^{2}$

The standard matrix of T which firstly reflects the points through vertical${x}_{2}-a\xi s$ and then reflects points through the line $x}_{2}={x}_{1$ .

Let us consider the standard basis of$R}^{2$ that is

$\{{e}_{1}=(1,0),{e}_{2}=(0,1)\}$

Step 2

Reflection of point along vertical${x}_{2}-a\xi s$ .

For any

$({x}_{1},{x}_{2})\in {R}^{2}$

$T({x}_{1},{x}_{2})=(-{x}_{1},{x}_{2})$

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$x}_{2}={x}_{1$ .

For any$({x}_{1},{x}_{2})\in {R}^{2}$

$Y({x}_{1},{x}_{2})=({x}_{2},{x}_{1})$

Now finding the images of obtained points under the reflection of points along$x}_{2}={x}_{1$ .

$T(-1,0)=(0,-1)$

$T(0,1)=(1,0)$

Therefore, the matrix of required linear transformation is given by

$$A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$$

It is given that T is linear transformation

The standard matrix of T which firstly reflects the points through vertical

Let us consider the standard basis of

Step 2

Reflection of point along vertical

For any

Firstly finding the images of standard basis under this transformation.

Step 3

Reflection points through the line

For any

Now finding the images of obtained points under the reflection of points along

Therefore, the matrix of required linear transformation is given by

Fasaniu

Beginner2021-12-19Added 46 answers

Step 1

We are looking for a $2\times 2$ matrix that reflects points through the horizontal axis and then reflects points through the line $x}_{2}={x}_{1$.

First the points need to be reflected through the horizontal ${x}_{1}-a\xi s$. This then means that the ${x}_{1}-coordinate$ remains unaffected and the ${x}_{2}-coordinate$ changes sign:

$${R}_{1}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

Note: if you multiply this matrix by the vector $({x}_{1},{x}_{2})}^{T$ to the right, then you obtain the vector $({x}_{1},-{x}_{2})}^{T$.

Step 2

Thus, the points need to be reflected through the line $x}_{2}={x}_{1$. The ${x}_{1}-coordinate$ and ${x}_{2}-coordinate$ then interchange.

$${R}_{2}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$

Note: if you multiply this matrix by the vector $({x}_{1},{x}_{2})}^{T$ to the right, then you obtain the vector $({x}_{2},{x}_{1})}^{T$.

Step 3

The result of the reflection matrices, with the first reflection matrix on the right, is then the combination of the reflections.

$$T={R}_{2}\times {R}_{1}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$$

Note: if you multiply this matrix by the vector $({x}_{1},{x}_{2})}^{T$ to the right then you obtain the vector $(-{x}_{2},{x}_{1})}^{T$

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