Conceptual Problems

Understanding the relation between solutions of system of linear equations, linear independence, and spanning

For each the following statements, answer: “always”, “sometimes”, or “never”, and provide a clear explanation that justifies your answer.

  1. Suppose that \(A\) is a \(3\times4\) matrix.
    1. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a solution.
    2. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a unique solution.
    3. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has infinitely many solutions.
  2. Suppose that \(A\) is a \(5\times4\) matrix.
    1. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a solution.
    2. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a unique solution.
    3. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has infinitely many solutions.
  3. Let \(A\) be a \(3\times 4\) matrix, with columns \(A=\begin{bmatrix}\boldsymbol{c}_1&\boldsymbol{c}_2&\boldsymbol{c}_3&\boldsymbol{c}_4\end{bmatrix}\) and with rows represented by the columns of the transpose \(A^T=\begin{bmatrix}\boldsymbol{r}_1&\boldsymbol{r}_2&\boldsymbol{r}_3\end{bmatrix}\)
    1. The columns of \(A\), \(\{\boldsymbol{c}_1,\boldsymbol{c}_2,\boldsymbol{c}_3,\boldsymbol{c}_4\}\), are linearly independent.
    2. The columns of \(A\), \(\{\boldsymbol{c}_1,\boldsymbol{c}_2,\boldsymbol{c}_3,\boldsymbol{c}_4\}\), span \(\mathbb{R}^3\)
    3. The columns of \(A^T\), \(\{\boldsymbol{r}_1,\boldsymbol{r}_2,\boldsymbol{r}_3\}\), are linearly independent.
    4. The columns of \(A^T\), \(\{\boldsymbol{r}_1,\boldsymbol{r}_2,\boldsymbol{r}_3\}\), span \(\mathbb{R}^4\)
  4. Let \(\boldsymbol{b}\in \mathbb{R}^4\) and let \(A\) be a \(4\times 4\) matrix, with columns \(A=\begin{bmatrix}\boldsymbol{c}_1& \boldsymbol{c}_2&\boldsymbol{c}_3&\boldsymbol{c}_4\end{bmatrix}\). Suppose that \(\{\boldsymbol{c}_1,\boldsymbol{c}_2,\boldsymbol{c}_3,\boldsymbol{c}_4\}\) is linearly independent.
    1. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a unique solution.
    2. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has infinitely many solutions.
    3. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a solution.
    4. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a unique solution (assume there is at least one solution).
    5. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has infinitely many solution (assume there is at least one solution).
    6. The columns \(\{\boldsymbol{c}_1,\boldsymbol{c}_2,\boldsymbol{c}_3,\boldsymbol{c}_4\}\) span \(\mathbb{R}^4\).
  5. Let \(\boldsymbol{b}\in \mathbb{R}^4\) and let \(A\) be a \(4\times 4\) matrix, with columns \(A=\begin{bmatrix}\boldsymbol{c}_1& \boldsymbol{c}_2&\boldsymbol{c}_3&\boldsymbol{c}_4\end{bmatrix}\). Suppose that \(\{\boldsymbol{c}_1,\boldsymbol{c}_2,\boldsymbol{c}_3,\boldsymbol{c}_4\}\) spans \(\mathbb{R}^4\).
    1. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a solution.
    2. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a unique solution (assume there is at least one solution).
    3. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has infinitely many solutions (assume there is at least one solution).
    4. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a solution.
    5. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a unique solution.
    6. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has infinitely many solutions.
    7. The columns \(\{\boldsymbol{c}_1,\boldsymbol{c}_2,\boldsymbol{c}_3,\boldsymbol{c}_4\}\) are linearly independence.
  6. Let \(\boldsymbol{b}\in \mathbb{R}^5\) and let \(A\) be a \(5\times 4\) matrix with columns \(A=\begin{bmatrix}\boldsymbol{c}_1& \boldsymbol{c}_2&\boldsymbol{c}_3&\boldsymbol{c}_4\end{bmatrix}\). Suppose that \(\{\boldsymbol{c}_1,\boldsymbol{c}_2,\boldsymbol{c}_3,\boldsymbol{c}_4\}\) is linearly independent.
    1. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a unique solution.
    2. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has infinitely many solutions.
    3. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a solution.
    4. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a unique solution (assume there is at least one solution).
    5. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has infinitely many solution (assume there is at least one solution).
  7. Let \(\boldsymbol{b}\in \mathbb{R}^4\) and let \(A\) be a \(4\times 5\) matrix with columns \(A=\begin{bmatrix}\boldsymbol{c}_1& \boldsymbol{c}_2&\boldsymbol{c}_3&\boldsymbol{c}_4&\boldsymbol{c}_5\end{bmatrix}\) and suppose that the row reduced matrix of \(A\) has exactly one free variable.
    1. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has a unique solution.
    2. The equation \(A\boldsymbol{x}=\boldsymbol{0}\) has infinitely many solutions.
    3. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a solution.
    4. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has a unique solution (assume there is at least one solution).
    5. The equation \(A\boldsymbol{x}=\boldsymbol{b}\) has infinitely many solution (assume there is at least one solution).