Instructor: Colin Grudzien

## Instructions: We will work through the following series of activities as a group and hold small group work and discussions in Zoom Breakout Rooms. Follow the instructions in each sub-section when the instructor assigns you to a breakout room. ## Activities: To perform some matrix operations, it can be helpful to use some non-base functions designed for this. We will require the library `matlab` ```{r} require("matlab") ``` ### Activity 1: Linear inverse problem #### Question 1: Suppose the matrix `A` from the last activity ```{r} A <- matrix(seq(3, 27, by=3), nrow=3, ncol=3) ``` defines a linear inverse problem for an unknown vector $\mathbf{x}$ and an observed vector, ```{r} b <- ones(3,1) ``` Is there any vector $\mathbf{x}$ that satisfies, $$\mathbf{A} \mathbf{x} = \mathbf{b}?$$ Why does such a choice exist or why not? #### Question 2: ```{r} b <- zeros(3,1) b ``` Is there any vector $\mathbf{x}$ that satisfies, $$\mathbf{A} \mathbf{x} = \mathbf{b}?$$ Why does such a choice exist or why not? #### Question 3: Set a random seed to be equal to zero and generate a 4-by-4 dimension matrix with entries given by a sample size $n=16$ from the standard normal. Suppose that this defines the relationship between the observed variable ```{r} b <- ones(4,1) ``` and an unknown vector $\mathbf{x}$ as, $$\mathbf{A}\mathbf{x}= \mathbf{b}.$$ Does any `x` as above exist? If one does find the solution. Explain why the solution exists or not due to the eigenvalues of the matrix. ### Activity 2: matrix norms Recall that matrix norms can be computed in R with the norm function. In the following we will consider some of their properties. #### Question 1: Compute the eigenvalues of the matrix `B` computed from matrix A below ```{r} B <- t(A) %*% A ``` then compute the sum of the square values. #### Question 2: Using the `norm` function, compute the norm-squared of the matrix `B`. Try changing the default norm -- which choice of norm-squared looks familiar?