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Math 221, Fall 2026

Problem Set 5: Subspaces, Bases, and Dimension

DueFriday, October 9th by 11:59pm

I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.

From Flatland by Edwin A. Abbott

Solutions must be typeset in LaTeX and written in complete, grammatical sentences. You may consult the course notes, the supplementary textbook, your classmates, your professor, math interns, or Q-Center tutors for help with the assignment, but all sources and collaborators must be explicitly referenced. Everyone is expected to write their own solutions. Feel free to DM on Discord if you have any questions. Problems marked (Challenge) are optional and are for extra credit

Problem 5.1

Let \[ A = \begin{bmatrix*}[r] 1 & 0 & 2 & -1 \\ 1 & 1 & 1 & 2 \\ 2 & -1 & 5 & -5 \end{bmatrix*}. \] You may use the Row Reducer to help you calculate.

  1. Find a basis for $\Nul(A)$. What is $\dim \Nul(A)$?
  2. Find a basis for $\col(A)$. What is $\dim \col(A)$?
  3. Is the vector $b = \begin{bmatrix*}[r] 2 \\ -1 \\ 7 \end{bmatrix*}$ in $\col(A)$?
  4. Is the vector $v = \begin{bmatrix*}[r] -3 \\ 4 \\ 1 \\ -1 \end{bmatrix*}$ in $\Nul(A)$?
  5. (Challenge) Find an equation in $b_1, b_2, b_3$ that is satisfied exactly when the vector $b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$ belongs to the column space $\col(A)$.

Problem 5.2

For each of the following collections of vectors, decide whether it is a basis of $\RR^3$. Explain your answer.

  1. \( \blue{ v_1 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix},\quad v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix},\quad v_3 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} } \)
  2. \( \blue{ v_1 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix},\quad v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix},\quad v_3 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix},\quad v_4 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} } \)
  3. \( \blue{ v_1 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix},\quad v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} } \)
  4. \( \blue{ v_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix},\quad v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix},\quad v_3 = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} } \)

Problem 5.3

The vectors \[ v_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix},\qquad v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix} \] are linearly independent.

  1. Explain why $v_1, v_2$ is not a basis of $\RR^4$.
  2. Find a basis of $\RR^4$ that includes $v_1$ and $v_2$.

Problem 5.4

Let $V = \sp\{v_1, v_2\}$, where \[ v_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix},\qquad v_2 = \begin{bmatrix*}[r] 1 \\ -1 \\ 2 \end{bmatrix*}, \] and let $\cB = \{v_1, v_2\}$.

  1. Explain why $\cB$ is a basis of $V$.
  2. The vector $w = \begin{bmatrix*}[r] 2 \\ 4 \\ -2 \end{bmatrix*}$ is in $V$. Find $[w]_\cB$.
  3. Is the vector $u = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ in $V$? If it is, find $[u]_\cB$.