MML - Review for Exam 2

We will have our second exam this Friday, February 21. 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 labeled either “Start Discussion” or “Continue Discussion”, depending on the status of those discussions. Following that link will take you to my forum where you can create an account with your UNCA email address. You can ask questions and/or post responses and answers there.

The problems

  1. Let \(U\) and \(V\) be subsets of \(\mathbb R^2\) defined by \[ U = \{\langle x,y \rangle \in \mathbb R^2: 2x+3y = 0\} \] and \[ V = \{\langle x,y \rangle \in \mathbb R^2: 2x+3y = 5\}. \] Which of these is a vector space? For the one that is not, explain clearly why.

  2. 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. Give a complete description of the column space of \(A\).
    2. Give a complete description of the null space of \(A\).
    3. Give a complete description of the range of \(A\).

  3. Show that the null space of a linear transformation is closed under linear combinations.

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

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

  5. Suppose that \(A\) and \(B\) are invertible matrices of the same size. Show that \((AB)^{-1} = B^{-1}A^{-1}\).

  6. 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.

  7. Let \(\langle \cdot,\cdot \rangle\) denote the inner product on \(\mathbb R^2\) defined by \[\langle \begin{bmatrix}x_1&x_2\end{bmatrix}^T, \begin{bmatrix}y_1&y_2\end{bmatrix}^T \rangle = 2x_1y_1 + 3 x_2y_2. \] Also let \(\mathbf{u} = \begin{bmatrix}1&2\end{bmatrix}^T\) and \(\mathbf{v} = \begin{bmatrix}-2&1\end{bmatrix}^T\).

    1. Compute \(\langle \begin{bmatrix}1&2\end{bmatrix}^T, \begin{bmatrix}-2&1\end{bmatrix}^T \rangle\).
    2. Are \(\mathbf{u}\) and \(\mathbf{v}\) orthogonal with respect to this inner product.
    3. Show that this inner product is distributive over vector addition.

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

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