Exam 1

This take-home exam is due at 11:00pm on Monday, 14th October 2024. Please write down the solutions neatly in your own handwriting, and submit the completed exam by email to amitra@mtech.edu. Please submit the exam as a single pdf file - you can scan / take a photo of your completed exam (handwritten) and convert it into a pdf file (please name your file first.last_exam1.pdf). To get credit, please provide all details and give complete reasoning for all your work. The maximum possible score for this exam is 30 points.

Exercise 1 (3 Points) Find a counterexample to the following statement:

“If \mathcal{B}=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4 \} is a basis of \mathbb{R}^4, and \mathbf{W} is a proper subspace of \mathbb{R}^4, then some subset of \mathcal{B} is a basis for \mathbf{W}”

Exercise 2 (3+3 Points) Let \mathbf{W} be the subset of all vectors in \mathbb{R}^3 whose last two components are equal, i.e. \mathbf{W}= \Big\{\begin{bmatrix}a \\ b \\ b\end{bmatrix}: a,b \in \mathbb{R}\Big\}

  1. Show that \mathbf{W} is a subspace of \mathbb{R}^3.
  2. Find two bases \mathcal{B}_1 and \mathcal{B}_2 for \mathbf{W}, such that there are no vectors that belong to both \mathcal{B}_1 and \mathcal{B}_2.

Exercise 3 (3 + 3 + 3 Points) Suppose n vectors from \mathbb{R}^m form the columns of a matrix \mathbf{A}.

  1. If the vectors are linearly independent, what is the rank of \mathbf{A}?

  2. If they span \mathbb{R}^m, what is the rank of \mathbf{A}?

  3. If they are a basis for \mathbb{R}^m, what can we say about m and n?

Give explanations for all your answers to get any credit.

Exercise 4 (4 Points) The first two columns of a 3 \times 3 matrix \mathbf{Q} are shown below. Add a third column so that the matrix is orthogonal. \mathbf{Q}=\begin{bmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{14}} &\quad \\ \frac{1}{\sqrt{3}} & \frac{2}{\sqrt{14}} &\quad \\ \frac{1}{\sqrt{3}} & -\frac{3}{\sqrt{14}} &\quad \end{bmatrix}

Exercise 5 (3+2 Points) Consider the vectors \mathbf{a}, \mathbf{b}, \mathbf{c} below: \mathbf{a}=\begin{bmatrix}0\\0\\1\end{bmatrix},\quad \mathbf{b}=\begin{bmatrix}0\\1\\1\end{bmatrix},\text{ and } \mathbf{c}=\begin{bmatrix}1\\1\\1\end{bmatrix}

  1. Apply the Gram-Schmidt process to find an orthonormal basis spanning the subspace spanned by \mathbf{a}, \mathbf{b}, \mathbf{c}. Show all the computations.
  2. Write the matrix \mathbf{A} (whose columns are \mathbf{a}, \mathbf{b}, \mathbf{c}) in the form \mathbf{A}=\mathbf{QR}.

Exercise 6 (3 Points) Show that the quadratic function of three variables p(x,y,z)=x^2+3y^2+z^2+2xy+2yz-2x+3z+2 has a unique global minimum and find that minimum value.