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More Practice Problems

Table of Contents

1. Intersection of Spans

Determine a vector with integer entries which appears in both of the following spans.

\begin{align*} \mathsf{span} \left\{ \begin{bmatrix}1 \\ 2 \\ -1\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix} \right\} \qquad \mathsf{span}\left\{ \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 3\end{bmatrix} \right\} \end{align*}

2. Dependence Relations

Suppose that \(\{\mathbf v_1, \mathbf v_2, \mathbf v_3\}\) is a linearly independent set of vectors and that

\begin{align*} \mathbf u_1 &= \mathbf v_1 + \mathbf v_3 \\ \mathbf u_2 &= -2 \mathbf v_1 + \mathbf v_2 \textcolor{red}{+} \mathbf v_3 \\ \mathbf u_3 &= -3 \mathbf v_1 - \mathbf v_2 - 6 \mathbf v_3 \end{align*}

Determine a dependence relation with integer weights for the vectors \(\{\mathbf u_1, \mathbf u_2, \mathbf u_3\}\).

3. True/False

Determine if each of the following statements are true or false. If it is false, give a counterexample.

  1. For any matrices \(A\) and \(B\), if \(AB = I\) then \(A\) is invertible and \(B = A^{-1}\).
  2. For any matrix \(A\) in \(\mathbb R^{10 \times 15}\) and any \(\mathbf b\) in \(\mathbb R^{10}\), the matrix equation \(A\mathbf x = \mathbf b\) has a solution.
  3. For any matrices \(A\) and \(B\), if \(AB = 0\), then \(A = 0\) or \(B = 0\).
  4. For any vectors \(\mathbf v_1\), \(\mathbf v_2\), \(\mathbf v_3\), if \(\mathbf v_1 \in \mathsf{span}\{\mathbf v_2, \mathbf v_3\}\) then \(\{ \mathbf v_1 + \mathbf v_2, \mathbf v_1 + \mathbf v_3\}\) is a linearly dependent set.
  5. For any matrices \(A\) and \(B\), if \(AB = BA\), then \(A = B\).

4. Inverses(?)

4.1. Multiply on the Right

Determine a matrix \(B\) with integer entries such that the following equality holds.

\begin{align*} \begin{bmatrix} \textcolor{red}{1} & -1 & 2 \\ -3 & 4 & 2 \end{bmatrix} B = I \end{align*}

4.2. Multiply on the Left

Explain why it is not possible to determine a matrix \(B\) such that the following equality holds.

\begin{align*} B \begin{bmatrix} \textcolor{red}{1} & -1 & 2 \\ -3 & 4 & 2 \end{bmatrix} = I \end{align*}

5. Linear Transformations

Suppose that \(T : \mathbb R^3 \to \mathbb R^3\) is the linear transformation which reflects vectors across the \(xy\) plane (i.e., across the plane given by the linear equation \(z = 0\)) and that \(S : \mathbb R^3 \to \mathbb R^3\) the transformation which rotates vectors around \(\mathsf{span}\{[1 \ \ 1 \ \ 0]^T\}\) by \(180\) degrees. Determine the matrix which implements \(S \circ T\), the composition of \(S\) and \(T\) (recall that \((S \circ T)(\mathbf v) = S(T(\mathbf v))\)).