Publications by Kristin Lussi
DATA 605 Week 4 Reponse
Define the linear transformation \(T: C^3 \rightarrow C^2\) \[ T( \begin{bmatrix} x_1\\ x_2\\ x_3\\ \end{bmatrix} ) = \begin{bmatrix} 2x_1 - x_2 + 5x_3\\ -4x_1 +2x_2 - 10x_3\\ \end{bmatrix} \] Let \(U,V\) be arbitrary vectors in \(C^3\). \[ u = \begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix}\\ v = \begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix} \] We must show...
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DATA 605: Week 4 Homework
Eigenshoes Load Libraries library(jpeg) library(imager) library(knitr) library(dplyr) files = list.files(path = '/Users/kristinlussi/Documents/MSDS/DATA 605/Week 4/jpg', pattern="\\.jpg") View Images # plot layout par(mfrow = c(4, 5), mar = c(0.5, 0.5, 0.5, 0.5)) for (file in files) { filepath <- file.path("/Users/kristinlussi/Documents/MSDS/D...
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DATA 605 Week 4 Discussion
Page 349 C40 If \(T: C^2 \rightarrow C^3\) satisfies \(T(\begin{bmatrix} 2\\ 1\\ \end{bmatrix}) = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\) & \(T(\begin{bmatrix} 1 \\ 1\\ \end{bmatrix}) = \begin{bmatrix} -1 \\ 2 \\ \end{bmatrix}\), find \(T(\begin{bmatrix} 4 \\ 3\\ \end{bmatrix})\). Solution Let’s express \(T(\begin{bmatrix} 4 \\ 3\\ \end{bmatrix})...
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DATA 605 Homework 3
Problem Set 1 What is the rank of matrix \(A\)? \[ A = \begin{bmatrix} 1 & 2 & 3 & 4\\ -1 & 0 & 1 & 3\\ 0 & 1 & -2 & 1\\ 5 & 4 & -2 & -3\\ \end{bmatrix} \] Solution by Hand Reduced Echelon Form: \[ \begin{bmatrix} 1 & 2 & 3 & 4\\ -1 & 0 & 1 & 3\\ 0 & 1 & -2 & 1\\ 5 & 4 & -2 & -3\\ \end{bmatrix} \rightarrow R2 = R1 + R2, R4 = R4-5R1 \rightarrow \b...
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DATA 605 Week 3 Discussion
C24 (Page 306) Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for \[ A = \begin{bmatrix} 1 & -1 & 1\\ -1 & 1 & -1\\ 1 & -1 & 1\\ \end{bmatrix}\\ \] Solution by Hand Find the eigenvalues \[ \det(A - \lambda I) = 0\\ \det( \begin{bmatrix} 1 & -1 & 1\\ -1 & 1 & -1\\ 1 & -1 & 1\\ \end{bmatrix} - \begin{bmatrix} \lambda & 0 ...
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DATA 605 Homework 2
Problem Set 1 Part 1: Proof & Demonstration Show that \[A^TA \neq AA^T \]in general. (Proof and demonstration.) Proof: Given a 2x2 matrix, \(A\): \[ A = \begin{bmatrix} a11 & a12 \\ a21 & a22 \\ \end{bmatrix} \] For any 2x2 matrix \(A\), the transpose of \(A\) is \[ A^T = \begin{bmatrix} a11 & a21 \\ a12 & a22 \\ \end{bmatrix} \] For any 2x2 matr...
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DATA 605 Week 2 Discussion
Exercise C25 (Page 278) Doing the computations by hand, find the determinant of the matrix below: \[ \begin{bmatrix} 3 & -1 & 4\\ 2 & 5 & 1\\ 2 & 0 & 6\\ \end{bmatrix} \] Define Matrix A = matrix(c(3, 2, 2, -1, 5, 0, 4, 1, 6), ncol = 3, nrow = 3) Solve with R det(A) ## [1] 60 Solve by Hand # 3 * det|5,1,0,6| determinant_A = (A[1,1] * ((A[2,2] *...
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DATA 605 Week 1 Discussion
Exercise C10 (Page 64): Compute: \[ 4 \begin{bmatrix} 2 \\ -3 \\ 4\\ 1\\ 0\\ \end{bmatrix} + (-2) \begin{bmatrix} 1\\ 2\\ -5\\ 2\\ 4\\ \end{bmatrix} + \begin{bmatrix} -1\\ 3\\ 0\\ 1\\ 2\\ \end{bmatrix} \] Solution # define matrices matrix_1 <- matrix(c(2,-3,4,1,0), ncol = 1) matrix_2 <- matrix(c(1,2,-5,2,4), ncol = 1) matrix_3 <- matrix(c(-1,3,0,1...
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DATA 605: Homework 1
Instructions For this assignment, build the first letters for both your first and last name using point plots in R. Then, write R code that will left multiply (%*%) a square matrix (x) against each of the vectors of points (y). Initially, that square matrix will be the Identity matrix. Use a loop that changes the transformation matrix incrementally...
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