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Chapter 2: Q2.7-3Q (page 93)

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

Translate by \(\left( {{\bf{3}},{\bf{1}}} \right)\), and then rotate \({\bf{45}}^\circ \) about the origin.

Short Answer

Expert verified

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

Step by step solution

01

Find the matrix for translation

The matrix for translation is \(\left( {3,1} \right)\).

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

02

Find the matrix for rotation

The matrix for rotation of \(45^\circ \) about the origin is

\(\left[ {\begin{array}{*{20}{c}}{\cos 45^\circ }&{ - \sin 45^\circ }&0\\{\sin 45^\circ }&{\cos 45^\circ }&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{ - \frac{1}{{\sqrt 2 }}}&0\\{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}&0\\0&0&1\end{array}} \right]\).

03

Find the combined matrix of transformation

The combined matrix for transformation can be expressed as shown below:

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

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

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Most popular questions from this chapter

How many rows does \(B\) have if \(BC\) is a \({\bf{3}} \times {\bf{4}}\) matrix?

Suppose Ais an \(m \times n\) matrix and there exist \(n \times m\) matrices C and D such that \(CA = {I_n}\) and \(AD = {I_m}\). Prove that \(m = n\) and \(C = D\). (Hint: Think about the product CAD.)

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\(A = \left( {\begin{aligned}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{aligned}} \right)\), \(C = \left( {\begin{aligned}{*{20}{c}}1&2\\{ - 2}&1\end{aligned}} \right)\), \(D = \left( {\begin{aligned}{*{20}{c}}3&5\\{ - 1}&4\end{aligned}} \right)\) and \(E = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\3\end{aligned}} \right)\)

\( - 2A\), \(B - 2A\), \(AC\), \(CD\).

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1鈥�4.

2. \[\left[ {\begin{array}{*{20}{c}}E&{\bf{0}}\\{\bf{0}}&F\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\]

In Exercises 27 and 28, view vectors in \({\mathbb{R}^n}\)as\(n \times 1\)matrices. For \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \) in \({\mathbb{R}^n}\), the matrix product \({{\mathop{\rm u}\nolimits} ^T}v\) is a \(1 \times 1\) matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product \({{\mathop{\rm uv}\nolimits} ^T}\) is a \(n \times n\) matrix, called the outer product of u and v. The products \({{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} \) and \({{\mathop{\rm uv}\nolimits} ^T}\) will appear later in the text.

28. If u and v are in \({\mathbb{R}^n}\), how are \({{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} \) and \({{\mathop{\rm v}\nolimits} ^T}{\mathop{\rm u}\nolimits} \) related? How are \({{\mathop{\rm uv}\nolimits} ^T}\) and \({\mathop{\rm v}\nolimits} {{\mathop{\rm u}\nolimits} ^T}\) related?
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