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MATH 1553
PRACTICE FINAL EXAMINATION
Name Section
1 2 3 4 5 6 7 8 9 10 Total
Please read all instructions carefully before beginning.
- The final exam is cumulative, covering all sections and topics on the master calendar.
- Each problem is worth 10 points. The maximum score on this exam is 100 points.
- You have 170 minutes to complete this exam.
- There are no aids of any kind (notes, text, etc.) allowed.
- Please show your work, unless instructed otherwise.
- You may cite any theorem proved in class or in the sections we covered in the text.
- Check your answers if you have time left! Most linear algebra computations can be easily verified for correctness.
- Good luck!
This is a practice exam. It is roughly similar in format, length, and difficulty to the real exam. It is not meant as a comprehensive list of study problems.
Problem 1. [2 points each]
In this problem, you need not explain your answers.
a) The matrix
0 0 1 0 0 1
is in reduced row echelon form:
- True 2. False b) How many solutions does the linear system corresponding to the augmented matrix 0 1 1 1 0 0 0 1
have?
- Zero.
- One.
- Infinity.
- Not enough information to determine.
c) Let T : R n^ → R m^ be a linear transformation with matrix A. Which of the following are equivalent to the statement that T is one-to-one? (Circle all that apply.)
- A has a pivot in each row.
- The columns of A are linearly independent.
- For all vectors v , w in R n , if T ( v ) = T ( w ) then v = w.
- A has n columns.
- Nul A = { 0 }.
d) Every square matrix has a (real or) complex eigenvalue.
- True 2. False e) Let A be an n × n matrix, and let T ( x ) = Ax be the associated matrix transforma- tion. Which of the following are equivalent to the statement that A is not invertible? (Circle all that apply.)
- There exists an n × n matrix B such that AB = 0.
- rank A = 0.
- det( A ) = 0.
- Nul A = { 0 }.
- There exist v 6 = w in R n^ such that T ( v ) = T ( w ).
Problem 3. [2 points each]
Short answer questions: you need not explain your answers.
a) Let A be an n × n matrix. Write the definition of an eigenvector and an eigenvalue of A.
b) Suppose u and v are orthogonal unit vectors, and let x = 2 u + v. Find ‖ x ‖.
c) Give an example of a 2 × 2 matrix that has no (real) eigenvectors.
d) Let W be the span of (1, 1, 1, 1) in R^4. Find a matrix whose null space is W ⊥.
e) Write a 3 × 3 matrix A with two (non-real) complex eigenvalues, whose eigenspace corresponding to λ = 7 is the x -axis.
Problem 4. [5 points each]
Let
A =
a) Compute A −^1 and det( A ). b) Solve for x in terms of the variables b 1 , b 2 , b 3 :
Ax =
b 1 b 2 b 3
Problem 6.
Suppose that your roomate Jamie is currently taking Math 1551. Jamie scored 72% on the first exam, 81% on the second exam, and 84% on the third exam. Not having taken linear algebra yet, Jamie does not know what kind of score to expect on the final exam. Luckily, you can help out.
a) [4 points] The general equation of a line in R^2 is y = C + Dx. Write down the system of linear equations in C and D that would be satisfied by a line passing through the points (1, 72), (2, 81), and (3, 84), and then write down the corresponding matrix equation. b) [4 points] Solve the corresponding least squares problem for C and D , and use this to write down and draw the the best fit line below.
70
80
90
1 2 3 4
y = + x
c) [2 points] What score does this line predict for the fourth (final) exam?
Problem 7.
Consider the vectors
v 1 =
v 2 =
v 3 =
v 4 =
and the subspace W = Span{ v 1 , v 2 , v 3 , v 4 }.
a) [2 points] Find a linear dependence relation among v 1 , v 2 , v 3 , v 4. b) [3 points] What is the dimension of W? c) [3 points] Which subsets of { v 1 , v 2 , v 3 , v 4 } form a basis for W? d) [2 points] Choose a basis B for W from (c), and find the B-coordinates of the vector w = (0, 0, 4, 0).
[Hint: it is helpful, but not necessary, to use the fact that { v 1 , v 2 , v 3 } is orthogonal.]
Problem 9.
Consider the matrix
A =
Å
ã .
a) [2 points] Compute the characteristic polynomial of A. b) [2 points] The complex number λ = 5 − 4 i is an eigenvalue of A. What is the other eigenvalue? Produce eigenvectors for both eigenvalues. c) [3 points] Find an invertible matrix P and a rotation-scaling matrix C such that A = PC P −^1. d) [1 point ] By what factor does C scale? e) [2 points] What ray does C rotate the positive x -axis onto? Draw it below.
rotate this
Problem 10.
Let L be a line through the origin in R^2. The reflection over L is the linear transformation ref L : R^2 → R^2 defined by
ref L ( x ) = x − 2 xL ⊥ = 2 proj L ( x ) − x.
a) [3 points] Draw (and label) ref L ( u ), ref L ( v ), and ref L ( w ) in the picture below. [Hint: think geometrically]
L
u
v (^) w
In what follows, L does not necessarily refer to the line pictured above. b) [2 points] If A is the matrix for ref L , what is A^2? c) [3 points] What are the eigenvalues and eigenspaces of A? d) [2 points] Is A diagonalizable? If so, what diagonal matrix is it similar to?
[Scratch work]
