Draft

Exam 1

Exercise 1 (10 Points) Using NumPy, crate a matrix of size 150\times3 whose columns contain random numbers generated using normal(0,1), uniform(-20,20), randInt(0,10), respectively. Then, append a column to it to hold the row sums.

Your code must not use any for or while loops.

Exercise 2 (10 Points) Show that for a square matrix of size n\times n, the determinant is zero if and only if the rank of the matrix is less than n.

Argue in both directions.

Exercise 3 (10 Points) Find a counterexample to the following statement:

If v_1, v_2, v_3, v_4 is a basis of \mathbb{R}^4, and W is a subspace, then some subset of the v’s is a basis for W.

Exercise 4 (10 Points) Find two different bases for the subspace of all vectors in \mathbb{R}^3 whose last two components are equal.

Exercise 5 (3 + 3 + 4 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 then?

Exercise 6 (10 Points) Find a third column so that the matrix \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} is orthogonal.

Exercise 7 (10 Points) Apply the Gram-Schmidt process to \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} Write the result in the form \mathbf{A}=\mathbf{QR}. Show all the computations.