Homework 4

Consider four vectors \(\mathbf v_1\), \(\mathbf v_2\), \(\mathbf v_3\) and \(\mathbf v_4\) in \(\mathbb R^4\) with the property that

\begin{align*} \begin{bmatrix} \mathbf v_1 & \mathbf v_2 & \mathbf v_3 & \mathbf v_4 \end{bmatrix} \sim \begin{bmatrix} 1 & 1 & 3 & -2 \\ 0 & 4 & -4 & 12 \\ 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} \end{align*}

For each of the following sets of vectors, determine the values of \(h\), if any, for which the set of vectors is linearly dependent. Show your work.

Consider the following set of vectors

\begin{align*} \mathbf v_1 = \begin{bmatrix} -10 \\ -6 \\ 2 \\ 0 \\ 8 \\ -3 \\ -4 \end{bmatrix} \qquad \mathbf v_2 = \begin{bmatrix} -3 \\ 3 \\ -8 \\ 5 \\ -2 \\ 7 \\ -3 \end{bmatrix} \qquad \mathbf v_3 = \begin{bmatrix} -20 \\ -12 \\ 4 \\ 0 \\ 16 \\ -6 \\ -8 \end{bmatrix} \qquad \mathbf v_4 = \begin{bmatrix} 6 \\ 1 \\ 3 \\ -1 \\ 10 \\ 7 \\ 4 \end{bmatrix} \qquad \mathbf v_5 = \begin{bmatrix} -2 \\ -10 \\ 10 \\ -5 \\ -3 \\ -10 \\ -5 \end{bmatrix} \qquad \mathbf v_6 = \begin{bmatrix} -2 \\ 3 \\ -9 \\ -2 \\ -9 \\ -2 \\ -7 \end{bmatrix} \qquad \mathbf v_7 = \begin{bmatrix} 13 \\ 25 \\ -22 \\ 13 \\ 24 \\ 41 \\ 15 \end{bmatrix} \end{align*}

(10 points) Let \(T\) be a linear transformation with the following input-output behavior. (Errata: 9/27 12:15PM, There was an error in the given numbers the updated value is in \(\textcolor{red}{\text{red}}\) below).

\begin{align*} T \left( \begin{bmatrix} -3 \\ 9 \\ 5 \\ \textcolor{red}{-4} \end{bmatrix} \right) = \begin{bmatrix} -4 \\ 3 \\ 4 \end{bmatrix} \qquad T \left( \begin{bmatrix} -9 \\ -5 \\ 0 \\ -7 \end{bmatrix} \right) = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} \qquad T \left( \begin{bmatrix} 9 \\ -1 \\ -2 \\ 4 \end{bmatrix} \right) = \begin{bmatrix} 3 \\ -3 \\ 7 \end{bmatrix} \end{align*}

Determine the value of

\begin{align*} T \left( \begin{bmatrix} 30 \\ 0 \\ -7 \\ 22 \end{bmatrix} \right) \end{align*}

Consider the following transformation.

\begin{align*} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \mapsto \begin{bmatrix} (x^3 + y^3 + z^3)^{1 / 3} \\ y + z \end{bmatrix} \end{align*}

(10 points) As we start to use NumPy more, it'll be useful to have a collection of predicates (i.e., functions which return Boolean values) for the problems we consider in this course. Please make sure you've read through the supplementary NumPy tutorial before getting started.

You're given starter code in the file hw04.py. Please do not change the name of this file when you submit it, nor the names of functions included in the starter code. You may add your own functions, but you are not expected to. You're only required to fill in the TODO items in the starter code.

You're required to implement the following functions. It may look like a large number of functions, but they will require very few lines of code.

• is_con_aug(a), assuming a represents the augmented matrix of a linear system, returns True if that system is consistent and False otherwise.
• is_con_mat_eq(A, b) returns True if the matrix equation \(A\mathbf v = \mathbf b\) has a solution, and False otherwise.
• in_span(v, A) returns True if \(v\) is in the span of the columns of \(A\), and False otherwise.
• num_pivots(A) returns the number of pivot positions in A.
• full_span(A) returns True if the columns of A have full span and False otherwise.
• lin_ind(A) returns True if the columns of A are linearly independent, and False otherwise.

See the started code for more information about each function. You may also want to look at the function numpy.nonzero(a) in the numpy documentation. Lastly, you don't need to deal with dimension mismatches. You may assume that inputs are well-formed.

You'll upload the single python file hw04.py to Gradescope with your implementations of TODO items in the code. As usual we provide a subset of the unit tests which you can run yourself in a file called tests.py. You can use the command python -m unittest in the terminal to run these tests.