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Chapter 2: Q2.7-6Q (page 93)

In Exercises 3-8, find the \({\bf{3}} \times {\bf{3}}\) matrices that produce the described composite 2D transformations, using homogenous coordinates.

Rotate points \({\bf{30}}^\circ \) and then reflect through the x-axis.

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&0\\{ - \frac{1}{2}}&{ - \frac{{\sqrt 3 }}{2}}&0\\0&0&1\end{array}} \right]\)

Step by step solution

01

Find the matrix for rotation

The rotation matrix can be written as

\(\left[ {\begin{array}{*{20}{c}}{\cos 30^\circ }&{ - \sin 30^\circ }&0\\{\sin 30^\circ }&{\cos 30^\circ }&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&0\\{\frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}&0\\0&0&1\end{array}} \right]\).

02

Find the matrix for reflection

The matrix for reflection is

\(\left[ {\begin{array}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&1\end{array}} \right]\).

03

Find the combined matrix of transformation

The combined matrix for transformation can be expressed as

\(\left[ {\begin{array}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&0\\{\frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&0\\{ - \frac{1}{2}}&{ - \frac{{\sqrt 3 }}{2}}&0\\0&0&1\end{array}} \right]\).

So, the transformed matrix is \(\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&0\\{ - \frac{1}{2}}&{ - \frac{{\sqrt 3 }}{2}}&0\\0&0&1\end{array}} \right]\).

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Most popular questions from this chapter

Prove the Theorem 3(d) i.e., \({\left( {AB} \right)^T} = {B^T}{A^T}\).

Show that \({I_n}A = A\) when \(A\) is \(m \times n\) matrix. (Hint: Use the (column) definition of \({I_n}A\).)

In exercise 5 and 6, compute the product \(AB\) in two ways: (a) by the definition, where \(A{b_{\bf{1}}}\) and \(A{b_{\bf{2}}}\) are computed separately, and (b) by the row-column rule for computing \(AB\).

\(A = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}&{ - {\bf{2}}}\\{ - {\bf{3}}}&{\bf{0}}\\{\bf{3}}&{\bf{5}}\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}\\{\bf{2}}&{ - {\bf{1}}}\end{aligned}} \right)\)

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

Give a formula for \({\left( {ABx} \right)^T}\), where \({\bf{x}}\) is a vector and \(A\) and \(B\) are matrices of appropriate sizes.
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