5. For each value of a in the setf32;31:9;31:8;32:1;32:2g, compute
the characteristic polynomial of A and the eigenvalues. In each case, create a graph of the characteristic
polynomial p(t) = det(A tI) for 0 t 3. If possible, construct all graphs on one coordinate system.
Describe how the graphs reveal the changes in the eigenvalues as a changes.
3. (5 points) Let =fb1;b2;b3gbe a basis for the vector space V. Find T(3b1 4b2) when T is a linear
transformation from V to V whose matrix relative to is: [T] =
4. (9 points) De ne T: IP2!IR3 by T(p) =
(a) Find the image under T of p(t) = 5 + 3t
(b) Show that T is a linear transformation.
(c) Find the matrix T relative to the basis 1;t;t2 for IP2 and the standard basis for IR3.
5. (6 points) Let T : IR2 ! IR2 be de ned by T(x) = Ax. Find a basis for for IR2 with the property
that [T] is diagonal.
6. Load the BU.jpg le provided on Blackboard, and run the following script. on Matlab1
img_in=double((rgb2gray(imread(‘BU.jpg’))));
image(img_in)
[V,D]=eig(cov(img_in))
image((V(:,530:640)*V(:,530:640)’*img_in’)’)
All matrices are in capital letters and bold. All vectors are in lower case and bold. All scalars are lower case and not
bolded.
yif you are not familiar with gsubmit, come to my o ce hours and I am happy to show you how it works. No email
submissions will be accepted
1You can run help cov for example on Matlab command line to nd more information about the function cov
(a) (1 point) What does the function cov in Matlab do?
(b) (3 points) What does cov(img in) do? Why is this required? What does the value of each cell in
this matrix indicate in context of img in?
(c) (1 point) What does the eig function in Matlab do?
(d) (3 points) What does the V and the D matrix in the above script. contain? i.e. what are the
columns of V? And what are the diagonal entries of D contain?
(e) (3 points) Why does the V 1 == VT. What is special about the columns of the matrix V? Hint:
V(:;i)TV(:;j) = 0 where i6= j and i and j are columns of the matrix V.
(f) (3 points) What does V(:,530:640)’*img in’ mean? You must use your knowledge of change of
basis matrix when answering this. Is the image in img in getting compressed with the above code?
(g) (3 points) What does the number from 530 to 640 indicate? Why are these numbers su cient to
recreate the original image? What happens if you change the 530 to 640? How would you explain
the image that gets generated in Matlab? Make sure to provide a screenshot of the image from
MATLAB.
(h) (3 points) What happens when you take the smallest 400 eigenvectors and perform. compression
and then uncompression. What image gets produced? Why does this image is not closer to the
true image? You must use your knowledge of variance when answering this question. Make sure to
provide a screenshot of the image from MATLAB.
(i) (3 points) What does V(:,530:640)*V(:,530:640)’*img in’ mean? You must use your knowl-
edge of change of basis matrix when answering this.