1. Find a real data set for your course projects. If you are having trouble finding a data set, then consider the data sets available at https://www.kaggle.com/datasets. Provide a link for your data set, and a brief description (4–5 sentences) of why you are interested in these data.

  2. Load your data set in your R script file. Clean, format, print summary statistics, and present a variety of exploratory plots. What are some population features you might be able to learn from your data set?

  3. Prove the following theorem. Let \(V\) be a vector space and \(B = \{u_1, \ldots, u_n\}\) be a subset of \(V\). Then \(B\) is a basis if and only if each \(v \in V\) can be expressed uniquely as

    \[ v = a_1 u_1 + \cdots + a_n u_n \]

    for some set of scalars \(\{a_1, \ldots, a_n\}\).

  4. Let

    \[ A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}. \]

    Is \(A\) diagonalizable? If so, find the eigenvalues and eigenvectors of \(A\).

  5. Let \(M \in \mathbb{R}^{p \times p}\) and \(b \in \mathbb{R}^p\). Then for any solution \(x \in \mathbb{R}^p\), the system of linear equations

    \[ Mx = b \]

    is said to be in row echelon form if the matrix \(M\) is upper triangular. Recall from your linear algebra prerequisite course a system of linear equations can always be re-expressed in row echelon form via a series of row operations. Write a function in R that takes as input \((M, b)\) where \(M\) is a \(p \times p\) matrix and \(b\) is a \(p\)-dimensional column vector and returns the row echelon form of the arguments \(M\) and \(b\). This can be done via the Gaussian elimination algorithm (https://en.wikipedia.org/wiki/Gaussian_elimination).

  6. Write an algorithm to solve for at least one solution to a row echelon form expression of a system of linear equations (if a solution exists). More specifically, assume you begin as in the previous question with an arbitrary system \(Mx = b\). Then passing this system through the function that you wrote for the previous question will produce the row echelon form of the system, \(\widetilde{M}x = \tilde{b}\). Your function for this question will be able to solve for at least one solution (if a solution exists) to the system \(\widetilde{M}x = \tilde{b}\), which you know is in row echelon form. This can be done via the back substitution algorithm (https://en.wikipedia.org/wiki/Triangular_matrix#Forward_and_back_substitution).

  7. Use the function you wrote for the previous question to write a function to determine an eigenvector associated with an eigenvalue of a given triangular matrix \(A \in \mathbb{R}^{p \times p}\). Your function should take as an input (A, lambda), where lambda is a user-supplied eigenvalue of \(A\), and return an eigenvector associated with lambda.