# Pre Exam Practice Questions

1. Arrange the following functions from least complex to most complex, also write which functions are equally complex (if applicable)

- $f(n)=n^2 -2n$
- $g(n) = 4^{2n}$
- $i(n) = (\log_4 n) +\sqrt{log_4 n}$
- $j(n) =2^{4n}$

2. Given $f(n) \in o(g(n))$, which of the following relationships are true:

a. $f(n)g(n) \in o(f(n))$

b. $f(n)g(n) \in \omega(f(n))$

c. $f(n)g(n) \in \Theta(f(n))$

d. Not enough information to establish the relationships above

3. If $f(n) \in O(g(n))$, which of the following relationships are true:

a. $f(n) \notin \omega(g(n))$

b. $g(n) \in \Omega(f(n))$

c. $f(n) + g(n) \in \Theta(g(n))$

d. a and b

e. b and c

f. a and c

g. a, b and c

h. None of the above

4. Which of the following set of basis vectors will always generate less than 2 dimensional spans?

a. $\{ \begin{bmatrix}a\\b \end{bmatrix}, \begin{bmatrix}b\\a\end{bmatrix}\}$

b. $\{ \begin{bmatrix}c\\-c \end{bmatrix}, \begin{bmatrix}b\\-b\end{bmatrix}\}$

d. $\{ \begin{bmatrix}-a\\b \end{bmatrix}, \begin{bmatrix}a\\-b\end{bmatrix}\}$

d. a and b

e. b and c

f. a and c

g. a, b and c

h. None of the above

5. Given $A = \begin{bmatrix}1 &-2 &0\\-1&2&1\\ 2 &3 &3\end{bmatrix}$, solve for $A^{-1}$

6. Given $A = \begin{bmatrix}1 &-2 &0\\-1&2&1\\ 2 &3 &3\end{bmatrix}$, solve for $\det({A^{-1}})$

7. Given a some scalar value $u$ and a 2x2 transformation matrix $B$. What is the determinant of $uB$ in terms of $\det(B)$?

Given a linear trasformation $f(\vec{v})$ with the transformation matrix $T = \begin{bmatrix}2 &-2\\2 &1\end{bmatrix}$, and two-dimensional vectors, $\vec{p}=\begin{bmatrix}2\\3\end{bmatrix},\vec{q}=\begin{bmatrix}-1\\2\end{bmatrix},\vec{r}=\begin{bmatrix}-1\\-2\end{bmatrix}$

8. Solve for $f(\vec{p})$
9. Solve for $f(\vec{q})$
10. Solve for $f(f(\vec{r}))$
11. Solve for the eigenvectors and eigenvalues of the transformation matrix $\begin{bmatrix}-1 & 2\\3 & 1 \\\end{bmatrix}$.
12. Solve for the eigenvectors and eigenvalues of the transformation matrix $\begin{bmatrix}1 & a\\a & 1 \\\end{bmatrix}$.
13. Find the pair with the highest dot product among this set of vectors: $\vec{s} = \begin{bmatrix} 1 \\ 1 \\ 0\end{bmatrix}$, $\vec{t} = \begin{bmatrix} 2 \\ 1 \\ 1\end{bmatrix}$, $\vec{u} = \begin{bmatrix} -1 \\ 1 \\ 0\end{bmatrix}$, $\vec{v} = \begin{bmatrix} 1 \\ 1 \\ 2\end{bmatrix}$,