Section 3.4

Why Do It This Way

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Problem 3.4.1

Determine the matrix associated with each of the three linear functions from R2\mathbb {R}^{2} to R2\mathbb {R}^{2} described below. That is, determine the aija_{ij}'s such that

f(p→)=f(x1x2)=(a11x1+a12x2a21x1+a22x2). f(\overrightarrow {p}) = f\begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} a_{11}x_{1}+a_{12}x_{2} \\ a_{21}x_{1}+a_{22}x_{2} \end{pmatrix}.
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Remark.
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The three linear functions in question, used throughout this section: rotation of points counterclockwise through an angle θ\theta; reflection of points about the xx-axis; and projection of points onto the line y=xy=x in a perpendicular manner.

The three linear transformations used throughout section 3.4The three linear transformations used throughout section 3.4

Problem 3.4.2

By using matrix multiplication, determine the linear function obtained by performing a rotation followed by a reflection (using the rotation and reflection of exercise 3.4.1).

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Problem 3.4.3

By using matrix multiplication, determine the linear function obtained by first performing a reflection, then a rotation, and finally a projection (using the reflection, rotation, and projection of exercise 3.4.1).

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