Exercises 6.1 - 6.4
Chapter 6. Linear Transformations
6.1 Matrices as Transformations
Exercise 6.1. (Linear Transformation: Rotation)
Make a function file with a function name reflect_pt to find the reflection of the point \((a,b)\) about the line through the origin of the \(xy\)-plane that makes an angle of \(\theta\,^{\circ}\) with the positive \(x\)-axis. Make \(a\), \(b\), and \(\theta\) the inputs to the function and the reflection point \((x,y)\) the output. Using this function, find the result of this function for \((a,b)=(1,3)\) and \(\theta=12\) by the following commands:
>> [x, y]=reflect_pt(1, 3, 12)
Solution.
%--- The following is the function file 'reflect_pt.m'. ---%
function [x, y]=reflect_pt(a, b, ang)
theta=ang*(pi/180);
T = [cos(2*theta) sin(2*theta); sin(2*theta) -cos(2*theta)];
result=T*[a b]';
x=result(1); y=result(2);
end
MATLAB results.
x =
2.1338
y =
-2.3339
Exercise 6.2. (Linear Transformation: Projection)
Consider successive rotations of \(\mathbb{R}^{3}\) by an angle \(\theta_{1}\) degree about the \(x\)-axis, then by an angle \(\theta_{2}\) degree about the \(y\)-axis, and then by an angle \(\theta_{3}\) degree about the \(z\)-axis. Make a function file named comp_Rot to find an appropriate axis and angle of rotation that achieves the same result in one rotation. Make \(\theta_{1}, \theta_{2},\) and \(\theta_{3}\) degrees the inputs to the function and the axis and angle of rotation the outputs. Give the output axis as a unit vector. You may make the standard matrix for the composition of the rotations by multiplication of the standard matrices for the rotations about the position \(x\)-, \(y\)-, and \(z\)-axes, respectively. Using the function comp_Rot, check the result of the following commands:
>> [L, ang]=comp_Rot(45, 45, 45)
Solution.
%--- The following is the function file 'comp_Rot.m'. ---%
function [L, ang] = comp_Rot(ang1, ang2, ang3)
% Angle as a degree.
angle=[ang1 ang2 ang3];
% Convert angles from degrees to radians.
theta=angle*(pi/180);
% Rotation about x-axis, y-axis, and z-axis.
Rx = [1 0 0;
0 cos(theta(1)) -sin(theta(1));
0 sin(theta(1)) cos(theta(1))];
Ry = [cos(theta(2)) 0 sin(theta(2));
0 1 0;
-sin(theta(2)) 0 cos(theta(2))];
Rz = [cos(theta(3)) -sin(theta(3)) 0;
sin(theta(3)) cos(theta(3)) 0;
0 0 1];
% Composition of the three rotation matrices R = Ry*Rx*Rz.
R = Rz*Ry*Rx;
% Find the axis of rotation of R,
% by finding the eigenvector of R
% corresponding to the eigenvalue lambda=1.
% Find a nonzero vector satisfying Rx = x.
L = null(eye(3) - R);
% Make the axis of rotation unit vector.
L = L/norm(L);
% Find the angle of rotation of R.
% Note that w=(-x2,x1,0) is
% orthogonal to the axis of rotation x=(x1,x2,x3).
w = [-L(2) L(1) 0]';
rot_theta = acos((dot(w, R*w))/((norm(w)*norm(R*w))));
ang = ((rot_theta)*(180/pi));
end
MATLAB results.
L =
0.3574
0.8629
0.3574
ang =
64.7368
6.2 Geometry of Linear Operators
Exercise 6.3. (Rotation as an Orthogonal Transformation)
Let
\[A = \begin{bmatrix} -\frac{3}{7} & -\frac{2}{7} & -\frac{6}{7}\\ \frac{6}{7} & -\frac{3}{7} & -\frac{2}{7}\\ -\frac{2}{7} & -\frac{6}{7} & \frac{3}{7} \end{bmatrix}.\]Show that \(A\) represents a rotation, and use Formulas \((16)\) and \((17)\) in Section \(6.2\) to find the axis and angle of rotation.
Solution.
A = [-3/7 -2/7 -6/7; 6/7 -3/7 -2/7; -2/7 -6/7 3/7]; format short;
% Check that A*A' is the identity matrix.
disp('A*A^T = '); disp(A*A')
% Although the off diagonal entries are not exactly all zeros,
% the scaling factor suggests that roundoff error prevents
% the computed matrix from being the identity matrix.
% You can see that the product is exactly the identity matrix,
% when the symbolic computation is used.
% Check that det(A)=1 to conclude that A is orthogonal.
disp('det(A) = '); disp(det(A))
% Since A*A'=I and det(A) = 1, A represents a rotation.
% By (16) of Section 6.2,
% find the angle of rotation
% and convert the angle from radians to degrees.
theta = acos((trace(A)-1)/2); ang = ((theta)*(180/pi));
disp('The angle of rotation in degrees is'); disp(ang);
% By (17) of Section 6.2, find the axis of rotation.
% The initial point at the origin.
e1 = [1 0 0]';
% v is along the axis of rotation.
v = (A+A'+((1-trace(A))*eye(3))) * e1;
disp('The axis of rotation passes through the point'); disp(v');
MATLAB results.
A*A^T =
1.0000 0.0000 0.0000
0.0000 1.0000 0.0000
0.0000 0.0000 1.0000
det(A) =
1.0000
The angle of rotation in degrees is
135.5847
The axis of rotation passes through the point
0.5714 0.5714 -1.1429
6.3 Kernel and Range
Exercise 6.4. (Invertible Matrix as a Bijective Linear Transformation)
Consider the matrix
\[A = \left[\begin{array}{rrrr} 3& \hspace{1mm} -5& \hspace{1mm} -2&\hspace{3mm} 2\\ -4 & 7 & 4 & 4 \\ 4 & -9 & -3 & 7 \\ 2 & -6 & -3 & 2 \end{array} \right].\]Referring to the Theorem 6.3.15 in Section 6.3, show that \(T_{A} : \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is onto in at least four different ways. You may use several related MATLAB commands.
Solution.
A = [3 -5 -2 2; -4 7 4 4; 4 -9 -3 7; 2 -6 -3 2];
format short;
% By (a) in Theorem 6.3.15, use the command rref of A.
rref(A)
% Since the reduced row echelon form of A is the identity matrix,
% the linear transformation T is onto.
% By (d) in Theorem 6.3.15, use the command null of A.
% Find a basis for the null space of A.
null(A, 'r')
% Since the null space contains only the zero vector,
% the linear transformation T is onto.
% By (g) in Theorem 6.3.15, use the command rank of A.
% Find the number of linearly independent columns of A.
rank(A)
% Since the column vectors of A are linearly independent,
% the linear transformation T is onto.
% By (i) in Theorem 6.3.15, use the command det of A.
% Find the determinant of A.
det(A)
% Since det(A) is nonzero,
% the linear transformation T is onto.
% By (j) in Theorem 6.3.15, use the command eig of A.
% Find the eigenvalues A.
eig(A)
% Since 0 is not an eigenvalue of A,
% the linear transformation T is onto.
6.4 Composition and Invertibility of Linear Transformations
Exercise 6.5. (Compositions of Linear Transformations)
Consider successive rotations of \(\mathbb{R}^{3}\) by \(30\,^{\circ}\) about the \(z\)-axis, then by \(60\,^{\circ}\) about the \(x\)-axis, and then by \(45\,^{\circ}\) about the \(y\)-axis. If it is desired to execute the three rotations by a single rotation about an appropriate axis, what axis and angle should be used?
Solution.
% Angle as a degree.
ang1=30; ang2=60; ang3=45;
% Convert angles from degrees to radians.
theta1=((ang1)*(pi/180));
theta2=((ang2)*(pi/180));
theta3=((ang3)*(pi/180));
format short;
% Rotation about z-axis with the angle 30.
Rz = [cos(theta1) -sin(theta1) 0; sin(theta1) cos(theta1) 0; 0 0 1];
% Rotation about x-axis with the angle 60.
Rx = [1 0 0; 0 cos(theta2) -sin(theta2); 0 sin(theta2) cos(theta2)];
% Rotation about y-axis with the angle 45.
Ry = [cos(theta3) 0 sin(theta3); 0 1 0; -sin(theta3) 0 cos(theta3)];
% Composition of the three rotation matrices R = Ry*Rx*Rz.
R = Ry*Rx*Rz;
% Find the axis of rotation of R,
% by finding the eigenvector of R corresponding to the eigenvalue lambda=1.
% Find a nonzero vector satisfying Rx = x.
x = null(eye(3) - R);
% Find the angle of rotation of R.
% Note that w=(-x2,x1,0) is orthogonal to the axis of rotation x=(x1,x2,x3).
w = [-x(2) x(1) 0]';
theta = acos((dot(w, R*w))/((norm(w)*norm(R*w))));
% Convert the angle from radians to degrees.
ang = ((theta)*(180/pi));
disp('The angle of rotation in degrees is'); disp(ang);
disp('The axis of rotation is'); disp(x');
MATLAB results.
The angle of rotation in degrees is
69.3559
The axis of rotation is
-0.9350 -0.3525 -0.0391
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