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MTH 215 Homework 6

MTH 215 — Intro to Linear Algebra

Kyle Monette
Spring 2026

Homework 6

Suggested Deadline: Wednesday, April 29 at 11:59pm

Last Possible Deadline: Friday, May 1 at 11:59pm

Your work must follow the homework policies and submission guidelines in the Syllabus and on Brightspace.

Written Section
  • Solve the problems in this document. Your work must be handwritten (not typed).
  • You need to show all your work. Correct answers without supporting work will not receive credit.
  • Using techniques, theorems, definitions, etc. from material not covered in this assignment will not receive credit.
  • Your work must be written on the pdf document provided. You can get another copy in class if needed.
  • Submit your work as a pdf on Brightspace in the assignment titled “Homework 6-Written (pdf)”.
  • Use the filename YourLastName.pdf in your submission.
Some parts of some questions may require the use of MATLAB. For such questions, download the template files (.m script files) from Brightspace and type your code there. Submit your .m files to the Brightspace assignment titled “Homework 6-Written (MATLAB)”. Your .m files must follow the guidelines in the Homework Module on Brightspace. Note: Some questions utilizing MATLAB still require you to write work or solutions in this document.
Edfinity Section
  • Solve the problems on the software Edfinity. Edfinity grades are computed at the semester's end.
  • You do not show your work for these problems; the software only takes the final answer.
  • This portion—and only this portion—is submitted entirely on Edfinity.

Both sections (in their respective formats and locations described above) are due:

Wednesday, April 29 at 11:59pm or Friday, May 1 at 11:59pm
Late work will not be considered!

For All Questions: If you perform row operations, you must explicitly state all operations used and show the updated matrix at each step.

  1. If possible, diagonalize \[ A = \mat{rrrr} 5 & -3 & 0 & 9 \\ 0 & 3 & 1 & -2 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \rix \] and show all your work. If it's not possible, explain why not by quoting a result from the notes.

    Note: You can use MATLAB to aid with row operations, but need to write the RREF's here.

  2. Define the vectors \[ \vec{x} \eq \mat{r} 2 \\ 5 \\ 1 \rix, \qquad \vec{y} \eq \mat{r} -1 \\ -4 \\ 6 \rix .\]

    1. Compute $\vec{x} \bullet \vec{y}$ (the inner product)
    2. Compute $\norm{\vec{x}}$ and $\norm{\vec{y}}$.
    3. Find the angle between $\vec{x}$ and $\vec{y}$, in degrees.
    4. Compute $\vec{x} \vec{y}^T$, which is called the outer product and can be found using normal matrix multiplication.

  3. An $n\times n$ matrix $Q$ is called orthogonal if the following two conditions hold:

    1. The norm of each column is $1$. That is, the columns are unit vectors.
    2. All columns are orthogonal to each other.

    For example, this matrix (with the columns indicated) \[ Q \eq \mat{ccc} \vec{q}_1 & \vec{q}_2 & \vec{q}_3 \rix \eq \mat{ccc} 1 / \sqrt{2} & -1 /\sqrt{2} & 0 \\ 1 / \sqrt{2} & 1 /\sqrt{2} & 0 \\ 0 & 0 & -1 \rix \] is orthogonal because 1) $\norm{\vec{q}_1} = \norm{\vec{q}_2} = \norm{\vec{q}_3} = 1$, and 2) $\vec{q}_1^T \vec{q}_2 = 0$, $\vec{q}_1^T \vec{q}_3 = 0$, and $\vec{q}_2^T \vec{q}_3 = 0$.

    1. For the matrix $Q$ above, compute $Q^T Q$. Show all steps.
    2. Explain why $Q$ must be invertible. Clearly justify your reasoning.
    3. Let $\vec{x} = \mat{r} 2 \\ 3 \\ 0 \rix$. Find $Q \vec{x}$ where $Q$ is defined above.
    4. Compute $\norm{\vec{x}}$ and $\norm{Q \vec{x}}$ using your answer from part (c).

  4. MATLAB Question

    Consider the matrix \[ A \eq \mat{rrrr} -6 & -3 & 6 & 1 \\ -1 & 2 & 1 & -6 \\ 3 & 6 & 3 & -2 \\ 6 & -3 & 6 & -1 \\ 2 & -1 & 2 & 3 \\ -3 & 6 & 3 & 2 \\ -2 & -1 & 2 & -3 \\ 1 & 2 & 1 & 6 \rix .\]

    1. In MATLAB, define a new matrix $Q$ formed by normalizing the columns of $A$ so they have norm $1$. You can compute the norm of a column vector x in MATLAB simply by norm(x). You do not have to write this matrix below.
    2. Compute $Q^T Q$ and $Q Q^T$ in MATLAB, and display these to the command window. You do not have to copy these matrices here. Briefly summarize your findings, including the sizes of these matrices. How do $Q^TQ$ and $Q Q^T$ differ?
    3. Let $\vec{y} = (1,2,3,4,5,6,7,8)^T$ be a vector in $\R^8$. In MATLAB, compute $\vec{p} = Q Q^T \vec{y}$. Then, form the augmented matrix $\mat{c|c} A & \vec{p} \rix$ and compute its rank using the rank function in MATLAB. Display this number to the command window, and write it below. Explain, in words, why this means $\vec{p}$ is in $\Col (A)$.

      Hint: There is something else you need to compute to fully, correctly answer this question.

    4. Let $\vec{z} = \vec{y} - \vec{p}$, and then compute $\vec{z}^T \vec{p}$ in MATLAB. Recall that the transpose operation is done by the apostrophe as in x'. Is $\vec{z}$ orthogonal to $\vec{p}$?