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Chapter 3: Q20P (page 142)
Show that the determinant of a unitary matrix is a complex number with absolute value=1. Hint: See proof of equation (7.11).
The determinant of the unitary matrix is unity.
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Question: For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using
6.
A particle is traveling along the line (x-3)/2=(y+1)/(-2)=z-1. Write the equation of its path in the form . Find the distance of closest approach of the particle to the origin (that is, the distance from the origin to the line). If t represents time, show that the time of closest approach is . Use this value to check your answer for the distance of closest approach. Hint: See Figure 5.3. If P is the point of closest approach, what is ?
The diagonals of a rhombus (four-sided figure with all sides of equal length) are perpendicular and bisect each other.
Let each of the following matrices represent an active transformation of vectors in ( x , y )plane (axes fixed, vector rotated or reflected). As in Example 3, show that each matrix is orthogonal, find its determinant and find its rotation angle, or find the line of reflection.
(a): As in problem 12,
linear?
(b): Is a linear operator?
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