Last Updated: February 11, 2024 - Added question about vector norms.
This homework is not turned in for credit. It is meant to assess your readiness for this course. Answers are provided, but please attempt to answer the problems independently. Remember: reading a solution is different from arriving at one yourself.
Let $f:[0,1]^n \rightarrow \mathbb R$.
Let $D = \left\{ (x, y) | y - x^2 = 0 \right\}$
Describe the following sets:
Let $A \in \mathbb R^{4\times 5}, B \in \mathbb R^{5 \times 3}$ and $C = AB$ be matrices. How many rows and columns do they each have?
Let $A$ and $B$ as above. What can be said about the $BA$?
The binomial coefficient is defined as
$$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$
Calculate $\binom{10}{3}$
Let $\Sigma$ be an $k \times m$ matrix. Suppose we may write $\Sigma$ in block-matrix form
$$ \Sigma = \begin{pmatrix} \Sigma_1 & \Sigma_2 \\ \Sigma_3 & \Sigma_4\end{pmatrix} $$
Suppose we know that $\Sigma_3 \in \mathbb R^{3\times5}$. How big are the other matrices?
Let $A$ be the matrix:
$$ A = \begin{pmatrix} 1 & 2 & 1 \\ 3 & 0 & -1 \\ 3 & 2 & 1 \end{pmatrix} $$
Let $A$ be the matrix:
$$ A = \begin{pmatrix} 1 & 3 & 1 \\ 0 & 0 & 1 \\ 1 & 3 & 2 \end{pmatrix} $$
Let $\mathbf v_1, \mathbf v_2, \mathbf v_3 \in \mathbb R^5$.