University College Birmingham Compute the Algebraic Values Linear Algebra Questions

Book:

https://math.berkeley.edu/~yonah/files/Linear%20Al…

Name: __________________
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UCSD ID #:____________________
Math 18 Summer 2021
2 hours
1. No materials and calculators are allowed during the exam.
2. Write your solutions clearly and (a) Indicate the number and letter of each question; (b) Present your answers in the same order they appear in
the exam; (c) Start each question on a new page.
3. Show all your work; no credit will be given for unsupported answers.
4. Your answers must be based on the material presented in class; no credit will be given otherwise.
5. We reserve the right to examine you in person on your answers to the problems in this exam.
1 0 1 


1
1. Let A  0 1 0 Show that A is diagonalizable, and write it as A  PP . Find P and the diagonal matrix  . You do


1 0 1 
1
not need to find P . (15 Points)
2.
a. Construct a
3  3 matrix that is invertible but not diagonalizable. (10 Points)
b. Construct a non-diagonal
3  3 matrix that is diagonalizable but not invertible. (5 Points)
3. Let A be diagonalizable. Prove that the determinant of A is equal to the product of its eigenvalues. (10 Points)
4.
a. Construct a
Points)
3  3 matrix A with no zero entry and a vector b  0 in R 3 such that b is not in the column space of A . (10
1
 
b. Construct a 3  3 matrix A with no zero entry such that the vector b  1 is in NulA . (10 Points)
 
1
 4 5 9 2 


5. Let A  6 5 1 12 . An echelon form of A is the matrix


 3 4 8 3
a. Find a basis for
1 2 6 5
0 1 5 6  .


0 0 0 0 
ColA (5 Points)
b. Find a basis for NulA (10 Points)
6.
3
 1 2 3




a. Find the orthogonal projection of b  1 onto the column space of A  1 4 3 (15 Points)
 


 5 
 1 2 3
b. Find all the least square solutions of
Ax  b . (10 Points)
Problems 1-8 (Note u-v=ufv = vt u)
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