MML - Review for Exam 2

We will have our second exam next Friday, February 20. This review sheet is genuinely meant to help you succeed on that exam.

Generally, I will expect solutions to the problems, as opposed to just answers. So, for example, if the answer to an optimization problem is \(y=5\), then the solution will consist of a clear explanation with correctly written supporting computations indicating why the answer is \(y=5\).

Note that there is a link at the bottom of this sheet that will take you to a forum topic where you can ask questions about the sheet.

The problems

  1. Write down definitions of the following:

  2. A bit of data on NBA players is shown in Table 1.

    1. What are the cases in the data table?
    2. Name one numerical variable.
    3. Name one nominal, categorical variable.
    4. Name one ordinal, categorical variable.
Table 1: NBA Players
first_name last_name team team_abbr position number height
Alex Abrines Thunder OKC Guard 8 78
Jaylen Adams Hawks ATL Guard 10 74
Steven Adams Thunder OKC Center 12 84
Bam Adebayo Heat MIA Center-Forward 13 82

  1. Consider the numeric data \(\{1, 1, 2, 4\}\).

    1. Write down the computation showing that the mean is \(2\).
    2. Write down the computation showing that the standard deviation is \(\sqrt{3/2}\).

  2. Compute the determinants of the following matrices and use that information to determine the singularity or non-singularity of each matrix.

    1. \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 1 \\ 1 & -1 & 0 \end{bmatrix} \]
    2. \[ B = \begin{bmatrix} 1 & 2 & 0 & 8 & -1 & 0 & 6 & 10 \\ 2 & 0 & -8 & -1 & -1 & -1 & 1 & 1 \\ 0 & 3 & 3 & 12 & 0 & -1 & 0 & 2 \\ 1 & 10 & -6 & -25 & -1 & 0 & -1 & -4 \\ 3 & 3 & -74 & 4 & 5 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 8 & 0 & -1 & 5 & 1 & 0 \\ 4 & 0 & 1 & -4 & 0 & 1 & 0 & 1 \end{bmatrix} \]
    3. \[ C = \begin{bmatrix} 1 & 0 & -4 & 2 & 1 & -9 & 1 & 3 \\ 0 & 1 & 2 & 1 & 0 & 0 & 17 & 0 \\ 0 & 0 & 1 & -4 & 0 & 0 & -1 & -1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & -1 & -72 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & -4 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \]

  3. Suppose that the matrix \(A\) and its reduced row echelon form \(R\) are \[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 3 & 5 & 7 & 9 \end{bmatrix} \quad R = \begin{bmatrix} 1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

    1. Provide a basis for the column space of \(A\).
    2. Provide a basis for the null space of \(A\).
    3. Provide a basis for the range of \(A\).

  4. Show that the null space of a linear transformation mapping \(\mathbb R^n \to \mathbb R^m\) is closed under linear combinations and is, therefore, a subspace of \(\mathbb R^n\).

  1. Let \[ A = \begin{bmatrix}1&2\\3&5\end{bmatrix}. \]

    1. Row reduce the augmented matrix \([A|I]\).
    2. What is \(A^{-1}\)?

  2. Suppose that \(A\) is a non-singular matrix. Show that \[(A^{-1})^{\mathsf{T}} = (A^{\mathsf{T}})^{-1}.\]

  3. Find a value of \(t\) such that \[ \mathbf{x} = \begin{bmatrix}1 \\ 1 \\ -1 \\ 2t\end{bmatrix} \quad \text{and} \quad \mathbf{y} = \begin{bmatrix}2 \\ 1 \\ 2 \\ 3\end{bmatrix}. \] are perpendicular.

  4. Find the projection \(\text{proj}_{\mathbf{b}}\mathbf{x}\) of the vector \(\mathbf{x}\) onto \(\mathbf{b}\), where \[ \mathbf{x} = \begin{bmatrix}1 \\ 0 \\ -1 \\ 2\end{bmatrix} \quad \text{and} \quad \mathbf{b} = \begin{bmatrix}2 \\ 1 \\ 0 \\ 2\end{bmatrix}. \]

  5. Find the least squares solution to the overdetermined linear system \[ \begin{bmatrix} 1&0\\0&-1\\1&1 \end{bmatrix} \begin{bmatrix} x_1\\x_2 \end{bmatrix} = \begin{bmatrix} 1\\0\\1 \end{bmatrix}. \]

Your questions and answers

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