Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has been a while since you took linear algebra, or because you did not learn some of these concepts Don t worry We will use the Discussion Forum on https://spaceuhedu to ask questions and help one another 1 Suppose A is a real m by n matrix and a What does it mean to say that a vector is a solution to Ax = b? b What does it mean to say that the system Ax = b is consistent? c What does it mean to say that a vector is a least squares solution to Ax = b? 2 Suppose A is a real m by n matrix and a What is the system of normal equations associated with the system Ax = b, and what purpose does this system serve? b Suppose is a least squares solution to Ax = b How is Ax related to the vector b? 3 Suppose A is a real m by n matrix and True or False If the statement is true, then give a reason If the statement is false, then give a counter example a The system Ax = b always has at least one solution b If the system Ax = 0 has a nontrivial solution, then Ax = b is inconsistent c A least squares solution to Ax = b is also a solution to Ax = b d A solution to Ax = b is also a least squares solution to Ax = b 2x1x2 2x3 2 4 Use elementary row operations to solve the linear system x1 2x2 2x3 1 x1 5x2 4x3 1 x1x2 2 5 Show that the system 2x1x2 1 is inconsistent 2x1x2 4 6 Suppose A is a real n by n matrix Describe the process of using elementary row operations to determine if A is invertible, and if it is, finding the inverse of A
1 2 0 7 Use elementary row operations to find the inverse of the matrix 1 2 0 1 0 2 8 Suppose A is a real m by n matrix a What is Col (the column space of sometimes also called the range space of )? b What is Nul (the null space of sometimes also called the kernel of )? c Two vectors, are orthogonal (perpendicular) with respect to the Euclidean dot product if and only if 0 Two subspaces, are orthogonal if and only if 0 for every and In this case, we write Prove that Nul Col d Suppose is a subspace of Describe the orthogonal complement of with respect to the Euclidean dot product e Prove that Nul is the orthogonal complement of Col 1 1 0 0 1 1 9 Let V span,, Give all of the equations that must be satisfied by a vector 1 0 1 0 1 1 with Euclidean length 1 that is orthogonal to V (with respect to the Euclidean dot product) 10 State the rank theorem 11 Suppose A is an n by n matrix Give at least 10 equivalent statements to is invertible 12 Suppose A is a 6 by 8 matrix, and dim Col (A T ) = 3 a dim Nul (A T ) = b dim Nul (A) = c Give the size of A T A d Give the size of AA T e dim Col (A) = f Show that Col A T A is a subset of Col (A T )
2 0 0 1 3 1 1 2 0 0 0 1 3 0 1 1 2 0 0 1 3 1 1 2 g Is it possible for A to be row equivalent to? 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 Suppose A and B are n by n matrices What does it mean to say that A and B are similar matrices? List some consequences of A and B being similar matrices? 14 True or False If the statement is true, then give a reason If the result is false, then give a counter example a If two matrices have the same trace, then they are similar matrices b Elementary row operations, applied to a matrix, do not change the eigenvalues of the matrix c Elementary row operations, applied to an augmented matrix, do not change the solution set to the corresponding system of linear equations 15 Suppose,,, is a subset of a Describe span,,, b What has to be true for,,, to be an orthonormal subset of? c Suppose,,, is an orthonormal subset of Form the matrix whose columns are,,, What is? 16 Suppose,, a How do you create proj (the projection of onto span)? b How do you create proj, (the projection of onto span, )? c Explain how to find two orthogonal vectors whose span is, d Explain how to find the angle between and, 1 1 17 Let v 1 and u 1 1 1 a Give projvu (the projection of onto span)
b Give the distance from u to 1 span 1 1 18 Suppose is a subspace of What does it mean to say that is a subspace of? 19 Show that H 2 3, R is a subspace of 20 P 3 is the set of all polynomials with real coefficients, of a single variable, of degree less 3 R than or equal to 3 Define the function T : P 3 P 3 via T (y) = (2x 2 1)y + y a Show that P 3 is a vector space b Show that T is a linear transformation c Give the matrix representation for T with respect to the standard basis {1, x, x 2, x 3 } of P 3 3 1 21 Find the eigenvalue and associated eigenspaces for the matrix 1 3 22 A is a real 3 by 3 matrix with eigenvalues -1/2, 1/2 and 0 Suppose E 1/2 1 span 1, 1 E 1/2 1 1 1 1 1 1/2 1/2 0 span 1 and E0 span 1, and 1 1 1 1/2 1/2 1 Give 1 0 1 1 0 0 1 1 the matrix A 2 3 2 23 The matrix A is similar to 0 4 1 0 0 1 a Give the eigenvalues of A b Give the trace of A c Give the determinant of A 24 Suppose you have the data,,,,,, a Describe the process of finding the line of best least squares fit for the data 1
b Suppose y = mx + b is the line of best least squares fit for the data given in part a How do you find the predicted value of y at a value x = c? 25 Consider the (x,y) data given by 1, 1, 1, 2, 2, 4 a Find the line of best least squares fit for this data b Use this line to predict the value of y when x = 15