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Chapter 1: Q23Q (page 1)

A Givens rotation is a linear transformation from \({\mathbb{R}^{\bf{n}}}\) to \({\mathbb{R}^{\bf{n}}}\) used in computer programs to create a zero entry in a vector (usually a column of matrix). The standard matrix of a given rotations in \({\mathbb{R}^{\bf{2}}}\) has the form

\(\left( {\begin{aligned}{*{20}{c}}a&{ - b}\\b&a\end{aligned}} \right)\), \({a^2} + {b^2} = 1\)

Find \(a\) and \(b\) such that \(\left( {\begin{aligned}{*{20}{c}}4\\3\end{aligned}} \right)\) is rotated into \(\left( {\begin{aligned}{*{20}{c}}5\\0\end{aligned}} \right)\).

Short Answer

Expert verified

\(a = \frac{4}{5}\) and \(b = - \frac{3}{5}\)

Step by step solution

01

Find the equation for rotation

As the rotation matrix is \(\left( {\begin{aligned}{*{20}{c}}a&{ - b}\\b&a\end{aligned}} \right)\) and \(\left( {\begin{aligned}{*{20}{c}}4\\3\end{aligned}} \right)\) is rotated through \(\left( {\begin{aligned}{*{20}{c}}5\\0\end{aligned}} \right)\),

\(\left( {\begin{aligned}{*{20}{c}}a&{ - b}\\b&a\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}4\\3\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}5\\0\end{aligned}} \right)\).

The above equation implies that \(4a - 3b = 5\) and \(3a + 4b = 0\).

02

Form the augmented matrix for \(a\) and \(b\)

The augmented matrix for the equation of \(a\) and \(b\) is \(\left( {\begin{aligned}{*{20}{c}}4&{ - 3}&5\\3&4&0\end{aligned}} \right)\).

03

Solve the augmented matrix using row operations

At row two, multiply row two by 4 and row one by 3 and subtract row one from row two, i.e., \({R_2} \to 4{R_2} - 3{R_1}\).

\(\left( {\begin{aligned}{*{20}{c}}4&{ - 3}&5\\{3 \times 4 - 4 \times 3}&{4 \times 4 - 3\left( { - 3} \right)}&{0 - 5 \times 3}\end{aligned}} \right)\)

After performing the row operations, the matrix becomes

\(\left( {\begin{aligned}{*{20}{c}}4&{ - 3}&5\\0&{25}&{ - 15}\end{aligned}} \right)\).

Divide row two by 25, i.e., \({R_2} \to \frac{{{R_2}}}{{25}}\).

\(\left( {\begin{aligned}{*{20}{c}}4&{ - 3}&5\\0&1&{ - \frac{3}{5}}\end{aligned}} \right)\)

04

Solve the augmented matrix using row operations

At row one, multiply row two by 3 and add it to row one, i.e., \({R_1} \to {R_1} + 3{R_2}\).

\(\left( {\begin{aligned}{*{20}{c}}{4 + 0}&{ - 3 + 3 \times 1}&{5 - 3 \times \frac{3}{5}}\\0&1&{ - \frac{3}{5}}\end{aligned}} \right)\)

After performing the row operation, the matrix becomes

\(\left( {\begin{aligned}{*{20}{c}}4&0&{\frac{{16}}{5}}\\0&1&{ - \frac{3}{5}}\end{aligned}} \right)\).

Divide row one by \(4\), i.e., \({R_1} \to \frac{{{R_1}}}{4}\).

\(\left( {\begin{aligned}{*{20}{c}}1&0&{\frac{4}{5}}\\0&1&{ - \frac{3}{5}}\end{aligned}} \right)\)

So, \(a = \frac{4}{5}\) and \(b = - \frac{3}{5}\).

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Most popular questions from this chapter

In Exercise 23 and 24, make each statement True or False. Justify each answer.

23.

a. Another notation for the vector \(\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\end{array}} \right]\) is \(\left[ {\begin{array}{*{20}{c}}{ - 4}&3\end{array}} \right]\).

b. The points in the plane corresponding to \(\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 5}\\2\end{array}} \right]\) lie on a line through the origin.

c. An example of a linear combination of vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) is the vector \(\frac{1}{2}{{\mathop{\rm v}\nolimits} _1}\).

d. The solution set of the linear system whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&b\end{array}} \right]\) is the same as the solution set of the equation\({{\mathop{\rm x}\nolimits} _1}{a_1} + {x_2}{a_2} + {x_3}{a_3} = b\).

e. The set Span \(\left\{ {u,v} \right\}\) is always visualized as a plane through the origin.


Consider two vectors v鈫�1 andv鈫�2in R3 that are not parallel.

Which vectors inlocalid="1668167992227" 鈩�3are linear combinations ofv鈫�1andv鈫�2? Describe the set of these vectors geometrically. Include a sketch in your answer.

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

16. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\)

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)

Question: There exists a 2x2 matrix such thatA[12]=[34].
See all solutions

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