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.

Math notation


  1. Let $f:[0,1]^n \rightarrow \mathbb R$.

    1. Give an example of an element of the function $f$’s domain.
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    2. Give an example of an element of the functions $f$’s range.
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    3. Given an example of such a function $f$.
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  2. Let $D = \left\{ (x, y) | y - x^2 = 0 \right\}$

    1. Give three examples of the elements in $D$
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    2. Plot the elements of $D$ in the $(x, y)$ plane
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  3. Describe the following sets:

    1. $\mathbb R^5$
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    2. $\mathbb Z$
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    3. $\mathbb Z \times \mathbb R$
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  4. 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?

  5. Let $A$ and $B$ as above. What can be said about the $BA$?

  6. The binomial coefficient is defined as

    $$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$

    Calculate $\binom{10}{3}$

  7. 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?

Linear Algebra


  1. Let $A$ be the matrix:

    $$ A = \begin{pmatrix} 1 & 2 & 1 \\ 3 & 0 & -1 \\ 3 & 2 & 1 \end{pmatrix} $$

    1. What is $\text{det}(A)$?
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    2. What is $\text{rank}(A)$?
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  2. Let $A$ be the matrix:

    $$ A = \begin{pmatrix} 1 & 3 & 1 \\ 0 & 0 & 1 \\ 1 & 3 & 2 \end{pmatrix} $$

    1. Is $A$ invertible?
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    2. What is $\det(A)$?
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    3. How many vectors span the row space of $A$? Can you give some examples?
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  3. Let $\mathbf v_1, \mathbf v_2, \mathbf v_3 \in \mathbb R^5$.

    1. What is the largest dimensionality of $\text{span}(\mathbf v_1, \mathbf v_2, \mathbf v_3)$
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    2. Let $A$ be the matrix with the $\mathbf v_i$ as columns. What is the largest value $\text{rank}(A)$ can have? What is the smallest dimensionality of the nullspace or kernel $\text{ker}(A)$ can have?
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