###### Preview Activity2.6.1

Suppose that we wish to describe the geometric operation that reflects 2-dimensional vectors in the horiztonal axis. For instance, Figure 2.6.1 illustrates how a vector $$\xvec$$ is reflected into the vector $$T(\xvec)\text{.}$$

1. If $$\xvec = \twovec{2}{4}\text{,}$$ what is the vector $$T(\xvec)\text{?}$$ Sketch the vectors $$\xvec$$ and $$T(\xvec)\text{.}$$

2. More generally, if $$\xvec=\twovec{x}{y}\text{,}$$ what is $$T(\xvec)\text{?}$$

3. Find the vectors $$T\left(\twovec{1}{0}\right)$$ and $$T\left(\twovec{0}{1}\right)\text{.}$$

4. Use your results to write the matrix $$A$$ so that $$T(\xvec) = A\xvec\text{.}$$ Then verify that $$T\left(\twovec{x}{y}\right)$$ agrees with what you found in part b.

5. Describe the transformation that results from composing $$T$$ with itself; that is, what is the transformation $$T\circ T\text{?}$$ Explain how matrix multiplication can be used to justify your response.

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