Group HW01

Directions

  1. All answers have to be handwritten using pen and physical paper. Scan your answers and then upload as a pdf file to Animospace. Complete solutions with a sufficient level of explanation are preferred over incomplete ones.
  2. Only one person from the group submits the answers. Make sure that the group agrees to who should submit the answers.
  3. The deadline is May 13, 2024 before 10:00 am.
  4. You are already part of a group and the collaboration is only allowed within your group. That is the extent of collaboration that is allowed.
  5. Each member of the group should submit an evaluation form for the group homework.
  6. Read and consult the late policy in the syllabus if you have a tendency to be late with respect to submissions.
  7. Every group member has a responsibility to oneself and to the other members of their group. Hold each other accountable and give credit where it is due.

Exercises

  1. (2 points) Refer to EMEA for this exercise. Write the system \((*)\) found in page 617 in two forms: in the form (1) found in page 546 and in the form \(\mathbf{Ax} = \mathbf{b}\).

  2. (2 points) Construct your own simple example where you provide two matrices \(\mathbf{A}\) and \(\mathbf{B}\) such that \(\mathbf{AB}=\mathbf{0}\) but neither \(\mathbf{A}\) nor \(\mathbf{B}\) is \(\mathbf{0}\).

  3. (3 points) Define a square matrix \(\mathbf{A}\) to be idempotent when \(\mathbf{A}\mathbf{A}=\mathbf{A}\). Prove that if \(\mathbf{AB}=\mathbf{A}\) and \(\mathbf{BA}=\mathbf{B}\), then \(\mathbf{A}\) has to be idempotent. Make sure you supply a reason for every step you make.

  4. A row operation was applied to \(\mathbf{B}_1=\begin{pmatrix}1 & 2 & 3 \\ 4 & -5 & 6 \\ 6 & 7 & 5 \\ 7 & 9 & 8 \end{pmatrix}\) to produce \(\mathbf{B}_2=\begin{pmatrix}1 & 2 & 3 \\ 0 & -13 & -6 \\ 6 & 7 & 5 \\ 7 & 9 & 8\end{pmatrix}\).

    1. (1 point) Describe what row operation was applied.
    2. (1 point) Is there only one such row operation? Explain.
    3. (1 point) Provide a square matrix \(\mathbf{E}\) such that \(\mathbf{EB}_1=\mathbf{B}_2\).

  5. Suppose that \(f(x) = ax^2 + bx +c\) is a quadratic polynomial whose graph passes through the points \((—1, 1)\), \((0,0)\), and \((1, 2)\).

    1. (2 points) Find a linear system satisfied by \(a\), \(b\), and \(c\).
    2. (1 point) Solve the linear system using any method to determine what the function \(f\) is.

  6. Let a rectangle have length \(x\) and width \(y\).

    1. (1 point) Define functions \(P(x,y)\) and \(A(x,y)\) so that they would represent the perimeter and area of a rectangle, respectively.
    2. (1 point) Compute the differentials \(dP\) and \(dA\) in terms of \(dx\) and \(dy\).