##### Exercise6

This exercise concerns matrix transformations called projections.

1. Consider the matrix transformation $$T:\real^2\to\real^2$$ that assigns to a vector $$\xvec$$ the closest vector on horizontal axis as illustrated in Figure 20. This transformation is called the projection onto the horizontal axis. You may imagine $$T(\xvec)$$ as the shadow cast by $$\xvec$$ from a flashlight far up on the positive $$y$$-axis.

Find the matrix that defines this matrix transformation $$T\text{.}$$

2. Find the matrix that defines projection on the vertical axis.

3. What is the result of composing the projection onto the horizontal axis with the projection onto the vertical axis?

4. Find the matrix that defines projection onto the line $$y=x\text{.}$$

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