The point lies on so it remains unaffected by the transformation
That is, so it is an eigenvector with eigenvalue 1
The point doesn't lie on our graph
Equation of the line perpendicular to the given line and passing through the given point is:
Solving and we get
And
So the intersection point is
The reflected point is one which lies on and which has same distance from as the point
Which is to say it is the point where (since we want a different point)
Along with so that
Thus,
So the reflected point is which is note is
Thus, is also an eigenvector which eigenvalue
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k>0) a) (2 marks) Show...
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k> 0) are eigenvectors of L a) (2 marks) Show that v1 and vz b) (1 mark) What is the eigenvalue corresponding to each eigenvector? (Hint: No need to calculate the characteristic polynomial or solve a matrix equation. Geometric reasoning should suffice to solve this problem. Drawing a diagram is recommended!) 3. Let L be the linear transformation on R2 that...
0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue 0 4 -1 1 5. Given, A--2 6 -11 L-2 8-3 1 has the characteristic polynomial p(λ)-(x + 2) (z-2)2(z-1) Find the corresponding eigenvector for each eigenvalue
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....