How to prep for the exam

About the exam:

• The exam is closed-book, closed-notes, closed everything, in the same room as our class for all students including extended-time students.
• You will need to put away everything (all devices, bags, laptop etc). You are advised to visit the facilities before starting, and to come early.
• Questions will be mostly multiple choice.
• Questions will be distributed on paper and answers expected on an answer sheet.
• You will need to turn in both the questions and answers.
• The material for the exam is everything from Modules 3 to 12, and parts of modules 13, but with an emphasis on the most important concepts from these modules.
• The normal time for the exam is 60 minutes. The extended time is 90 minutes. Everyone will get 5 additional minutes to collect stuff together and turn in BOTH the answers and the exam questions.

How to prep:

• Review your own solutions to the non-programming module exercises. A slightly tweaked module exercise is fair game for the exam.
• Review at least the core concepts from:
  • Module 3: sections 3.2-3.8
  • Module 4: 4.1-4.7, 4.10, 4.12, 4.13
  • Module 5: 5.1-5.6
  • Module 6: 6.1-6.6
  • Module 7: 7.0-7.3, 7.6-7.7
  • Module 8: The definitions in Section 8.1, the meaning of the statements in each of Theorems 8.1, 8.2, 8.3.
  • Module 9: 9.1, 9.3, 9.5, 9.6
  • Review of core concepts.
  • Module 10: no questions from Module 10
  • Module 11: 11.0-11.2
  • Module 12: 12.0-12.3
  • Module 13: 13.0, 13.2, 13.4, 13.7
• Be familiar with definitions and the statements (but not necessarily the proofs) of theorems.

Sample questions

One set of sample questions are the non-programming module exercises, or slightly modified versions of them.

Here are some others:

1. Consider the following assertions:
   1. It is possible to add vectors with different numbers of elements as long as the larger one has zeroes where the smaller one has no elements, as in \((1,1,1,0,0)\) and \((2,2,2)\).
   2. The dot product of two vectors is zero only if at least one of them is the zero vector.
   Choose one of the following as the best answer:
   1. I and II are both true.
   2. I and II are both false.
   3. I is true but II is false.
   4. II is true but I is false.
   
   

2. Suppose that, in the equation \({\bf Ax}={\bf b}\), the matrix \({\bf A}\) is $$ {\bf A} \eql \mat{1 & 0 & 1\\ 1 & 1 & 2\\ 2 & 1 & 3} $$ and that \({\bf b}=(-1,0,-1)\). Consider the following assertions
   1. Because the third column of A is the sum of the first two columns, there is no solution.
   2. Because one equation has a zero as the coefficient of \(x_2\), the value of \(x_2\) can be anything.
   Choose one of the following as the best answer:
   1. I and II are both true.
   2. I and II are both false.
   3. I is true but II is false.
   4. II is true but I is false.
   
   

3. Consider an instance of \({\bf Ax}={\bf b}\) with 6 variables and 4 equations, and the following assertions:
   1. Because there are more variables than equations, there can be no solution.
   2. The RREF can have at most 4 pivots.
   Choose one of the following as the best answer:
   1. I and II are both true.
   2. I and II are both false.
   3. I is true but II is false.
   4. II is true but I is false.
   
   

4. Consider the following assertions:
   1. The product of the number 0 times the vector \({\bf 0}\) is the number 0.
   2. The product of the number 0 times the vector \({\bf 0}\) is the vector \({\bf 0}\).
   3. The dot product of the vector \({\bf 0}\) with any vector x is the number 0.
   4. The dot product of the vector \({\bf 0}\) with any vector x is the vector \({\bf 0}\).
   The only true assertions are:
   1. I and IV.
   2. I and III.
   3. II and III.
   4. II and IV.
   
   

5. If the matrix \({\bf A}\) rotates a vector clockwise by 45, and matrix \({\bf B}\) reflects about the y axis, then \({\bf BA}\) takes the vector (1,0) to
   1. \((-1,1)\)
   2. \((-1,-1)\)
   3. \((-1/\sqrt{2} , 1/\sqrt{2})\)
   4. \((-1/\sqrt{2} , -1/\sqrt{2})\)
   
   

6. Consider 3D space, the vectors \({\bf u}=(2,3,0), {\bf v}=(4,6,0), {\bf w}=(2,2,0)\) and these assertions:
   1. \({\bf u}\) and \({\bf v}\) are a basis for the xy-plane.
   2. \({\bf u}\) and \({\bf w}\) are a basis for the xy-plane.
   Choose one of the following as the best answer:
   1. I is true but II is false.
   2. II is true but I is false.
   3. I and II are both true.
   4. I and II are both false.
   
   

7. Consider the following proof of the statement: if \({\bf A}^{-1}\) exists then \({\bf x}= {\bf A}^{-1}{\bf b}\) is the unique solution to \({\bf Ax}={\bf b}\).
   1. \({\bf A}^{-1}\) is unique for any invertible matrix \({\bf A}\).
   2. Since we are given \({\bf b}\), there is only one possible product \({\bf A}^{-1}{\bf b}\).
   3. Since \({\bf x}= {\bf A}^{-1}{\bf b}\), the value of \({\bf x}\) is completely determined, and so, there is only one possible solution \({\bf x}\) to \({\bf Ax}={\bf b}\).
   Choose the best possible answer from:
   1. The proof is correct.
   2. Not correct because 1 is false
   3. Not correct because even though \({\bf A}^{-1}{\bf b}\) is a solution, there may be other solutions \({\bf y}\) where \({\bf Ay}={\bf b}\).
   4. Not correct because multiple solutions may exist if \({\bf A}\) is not square.
   
   

8. Suppose \({\bf u}, {\bf v}\) and \({\bf w}\) are orthogonal vectors. Consider the assertions
   1. Each of them has unit length.
   2. \(({\bf u} \cdot {\bf v}) \cdot {\bf w} = {\bf u} \cdot ({\bf v} \cdot {\bf w})\)
   Choose one of the following as the best answer:
   1. I and II are both true.
   2. I and II are both false.
   3. I is true but II is false.
   4. II is true but I is false.
   
   

9. Consider the following reasoning related to the least squares solution \( \hat{\bf x} = ({\bf A}^T {\bf A})^{-1} {\bf A}^T {\bf b} \)
   1. \( ({\bf A}^T {\bf A})^{-1} = {\bf A}^{-1} ({\bf A}^T)^{-1}\).
   2. Therefore, \( ({\bf A}^T {\bf A})^{-1} {\bf A}^T {\bf b} = {\bf A}^{-1} ({\bf A}^T)^{-1} {\bf A}^T {\bf b}\).
   3. Therefore \(\hat{\bf x} = {\bf A}^{-1} {\bf b}\).
   Choose the best answer:
   1. I is true for all matrices \({\bf A}\).
   2. I is true if \({\bf A}\) has an inverse.
   3. II follows from the properties of a matrix transpose.
   4. III demonstrates that the least squares solution is one way to prove that the inverse exists.
   
   

10. Consider any two non-collinear vectors \({\bf w}\) and \({\bf v}\) and let \({\bf y}=\mbox{proj}_{\bf v}({\bf w})\) be the projection of of \({\bf w}\) on \({\bf v}\). Consider these assertions:
    1. The angle between \({\bf w}\) and \({\bf y}\) is \(90^\circ\).
    2. The length of \({\bf y}\) is always less than that of \({\bf v}\).
    Choose one of the following as the best answer:
    1. I and II are both true.
    2. I and II are both false.
    3. I is true but II is false.
    4. II is true but I is false.
    
    

11. Suppose \({\bf x}_1,{\bf x}_2,\ldots,{\bf x}_n\) is a collection of orthogonal vectors in \(\mathbb{R}^n\). Let \({\bf v}\) be a vector and \({\bf w}=(\alpha_1,\alpha_2,\ldots,\alpha_n)\) be the coordinates of \({\bf v}\) using the \({\bf x}_i\)'s as a basis. Then, which of the following (perhaps more than one) are true? Explain your reasoning.
    1. \(\alpha_i = ({\bf v} \cdot {\bf x}_i) / ({\bf x}_i \cdot {\bf x}_i)\)
    2. \({\bf w} = {\bf A}^T {\bf v}\)
    3. \({\bf w} = {\bf A}^{-1} {\bf v}\)
    4. \(\alpha_i = |{\bf v}| |{\bf x}_i| \cos \theta_i\), where \(\theta_i\) is the angle between \({\bf v}\) and \({\bf x}_i\)
    
    

12. Suppose \({\bf A}_{m \times n}\) is a matrix and \({\bf x}\) is a nonzero vector such that \({\bf Ax}={\bf 0}\). Let \({\bf S}\) be the set of all vectors \({\bf y}\) such that \({\bf Ay}={\bf 0}\). Let \(r\) be the rank of \({\bf A}\). Then which of the following (possibly more than one) is true?
    1. \({\bf x}\) has m elements.
    2. \({\bf x}\) is in the nullspace.
    3. \(RREF({\bf A})\) has r pivots.
    4. A basis for \({\bf S}\) needs \(m-r\) vectors.
    
    

13. Suppose \(T\) is a linear transformation. Consider the following reasoning steps:
    1. \(T({\bf 0}) = T(1 \times {\bf 0} + (-1) \times {\bf 0})\)
    2. Next, \(T(1 \times {\bf 0} + (-1) \times {\bf 0}) = 1\times T({\bf 0}) - 1\times T({\bf 0})\)
    3. \(1 \times T({\bf 0}) - 1 \times T({\bf 0}) = {\bf 0}\)
    4. Therefore \(T({\bf 0}) = {\bf 0}\)
    Which of the following (possibly more than one) is true?
    1. I is true because we've applied the definition of a linear transformation.
    2. II is true because of the unique properties of \(T({\bf 0})\).
    3. III follows from the rules of integer arithmetic.
    4. IV concludes the result of reasoning in steps I-III.
    
    

14. Consider the following functions: \(f_1(t)=5, f_2(t)=2t^2, f_3(t)=1-t, f_4(t)=t^{-1}\). Let \(g(t) = 1 - t^2\) and \(h(t) = t^2 - t - 1\). Which of the following (possibly more than one) are true?
    1. \(g(t)\) can be written as a linear combination of \(f_1, f_2, f_3, f_4\).
    2. \(g(t)\) is a polynomial.
    3. \(h(t)\) can be written as a linear combination of \(f_1, f_2, f_3\).
    4. \(f_1, f_2\) form a basis for all polynomials of degree 2 or less.
    
    

15. Suppose \({\bf x}\) is an eigenvector of a matrix \({\bf A}\). Then which of the following (possibly more than one) are true?
    1. \({\bf Ax} = {\bf x}\).
    2. \(-{\bf x}\) is an eigenvector of \({\bf A}\).
    3. \({\bf x}\) is orthogonal to every column of \({\bf A}\).
    4. \({\bf A}\) is a square matrix.
    
    

16. Suppose \({\bf X}_{m\times n}\) is a data matrix (the data samples as columns) and \({\bf Y}\) is the same data in PCA coordinates. Consider these assertions:
    1. The purpose of PCA is to reduce variance in \({\bf Y}\) so that there's less noise in the transformed data.
    2. The matrix \({\bf S}^T\) that transforms into \({\bf X}\) to \({\bf Y}\) is an \(n \times m\) matrix.
    Choose one of the following as the best answer:
    1. I and II are both true.
    2. I and II are both false.
    3. I is true but II is false.
    4. II is true but I is false.