Assignment #6: Subspaces of R n, Bases, Dimension and Rank. Due date: Wednesday, October 26, 2016 (9:10am) Name: Section Number

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1 Assignment #6: Subspaces of R n, Bases, Dimension and Rank Due date: Wednesday, October 26, 206 (9:0am) Name: Section Number

2 Assignment #6: Subspaces of R n, Bases, Dimension and Rank Due date: Wednesday, October 26, 206 (9:0am) For full credit you must show all of your work.. (6 points) In each case, prove that the described set of vectors either does or doesn t form a subspace of R 3. 3x a. The set of vectors v = 2x, where x R.! x $ # & b. The set of vectors v = # x 2 &, where x R. # " x 3 & % x c. The set of vectors v = y, where x, y R. x y 2. (6 points) a. True or False: R 2 is a subspace of R 3. b. True or False: Any plane in R 3 is a subspace of R 3. c. True or False: A subspace can be the empty set. 3. (6 points) Let A = a. Is v = b. Is v = c. Is v = in the column space of A? How do you know? in the row space of A? How do you know? in the null space of A? How do you know?

3 4. (2 points) Let A = a. Give a basis for the column space of A. b. Give a basis for the row space of A. c. Give a basis for the null space of A. 5. (6 points) Find a basis for the set of vectors in R 3 in the plane x + 2y z = (2 points) Let A be an arbitrary m x n matrix, and let B be an arbitrary n x p matrix. a. Show that any vector x that is in the column space of AB is also in the column space of A. b. Is it also true that any vector x that is in the column space of AB is also in the column space of B? Why or why not? c. Is it also true that any vector x that is in the column space of A is also in the column space of AB? Why or why not? 7. (5 points) Let A be an arbitrary m x n matrix, and let B be an arbitrary n x p matrix. Show that any vector x that is in the null space of B is also in the null space of AB. 8. (3 points) What is the dimension of the subspace H of R 2 that is spanned by the three vectors 3 2 7,, 9 6 2? 9. (4 points) Suppose that A = row equivalent. a. What is the dimension of the column space of A? b. What is the dimension of the row space of A? c. What is the dimension of the null space of A? d. What is the rank of A? e. Give a basis for the column space of A. f. Give a basis for the row space of A. g. Give a basis for the null space of A. and B = (8 points) Let A be an m n non-zero matrix. For the questions below, please consider the possibilities: m > n, m < n and m = n. a. What is an upper bound on the rank of A? b. What is a lower bound on the rank of A? c. What is an upper bound on the dimension of the null space of A? d. What is a lower bound on the dimension of the null space of A? are

4 . (2 points) For each statement below, indicate whether it is True or False and briefly say why: a. If A is a 5 7 matrix with 3 pivot columns, cola = R 3. b. If A is a 5 7 matrix with 3 pivot columns, nula R 4. c. The rank of A T is always equal to the rank of A. d. The dimension of the null space of A T is always equal to the dimension of the null space of A. e. If A is an m n matrix, ranka + dim(nula) = n. f. If A is an m n matrix, ranka + dim(nula T ) = m. 2. (0 points) Some graphics applications implement 2D image rotation using a series of three shear operations, because shearing is quick and easy to accomplish on a raster image: a shear transformation simply shifts all of the pixels in each row of an image to the right or left by a different specified amount. In fact, it can be shown that: " $ # cosθ sinθ sinθ % " ' = $ cosθ & # a 0 %" ' $ &# 0 b %" ' $ &# a 0 % & ' where a = tan " θ % $ # 2 ' and b = sinθ. & Use Matlab to visualize the sequence of shear transformations that accomplish a rotation by θ = π 4. You can start out by loading and displaying a standard image from the Matlab library. For example: a = imread('coins.png'); figure; imshow(a); Next, you will need to define each of the two different shear operations described in the equation above, but using homogeneous coordinates (so that the matrices are 3 x 3 rather than 2 x 2), and use it to define a spatial transformation structure that can be applied to data of type image. For example: shear = [ -tan(pi/8) 0; 0 0; 0 0 ]; T = maketform('affine', shear); Once you have defined all of the needed transformations, your can apply them to your image using the function imtransform and look at the results. Calling figure again causes the new results to be displayed in a new window. For example: b = imtransform(a, T); figure; imshow(b); Please produce a series of 4 images, starting with the original one, and ending with what looks like a rotated version of the original one, with the two intermediate steps visible in between. Attach all of the images to your report, along with a copy of the commands you used to obtain them. Compare the results you obtain with this approach to the results you get by directly applying a single rotation transformation to the original image. What similarities and differences do you notice?

5 3. Extra credit: (2 points) Write a function in Matlab that takes as input an arbitrary matrix A and if it is possible to do so without pivoting, computes its LU factorization following the simple algorithm described in class and in your textbook. When the LU factorization cannot be found, the program gracefully returns L = I and U = A. You may not use Matlab s built-in function lu(). This question is asking you to implement your own simplified version of Matlab s lu() function. Please turn in your code via Moodle. The TAs will check your code by running it on the matrices in questions -3 above, as well as others for which an LU decomposition does not exist. To earn full credit, you are advised to test your code thoroughly. function [ l, u ] = my_lu( a ) %this function accepts an input matrix a and when possible produces as output %a matrix l that is unit lower triangular and a matrix u that is upper %triangular such that l*u = a end