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Problem 6. (10 pts.) Without first computing A-1, find A-1B where 1 0 A= ſi 0...
(33 pts) This question is about the matrix = ſi 2 [3 2 0 4 1...
(33 pts) This question is about the matrix = ſi 2 [3 2 0 4 1 6 3 1] 4 9 co (a) Find a lower triangular L and an upper triangular U so that A = LU. (b) Find the reduced row echelon form R = rref(A). How many independent columns in A? (c) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b
Find her Find ker (A) where A = where A-ſi 1 1] to . To write...
Find her Find ker (A) where A = where A-ſi 1 1] to . To write your answer properly, find vectors V1, ... , Vk so that ker(A) = span(V1, ..., Vk). 1 2 3 Let v1 = 1 , V2 = 2, V3 = 151. Determine if these vectors are 4 linearly independent. If they are not, then identify all redundant vectors.
T-1 2 5 ſi 0 0] Q2. (19 pts) Let B 0 1 8 and C...
T-1 2 5 ſi 0 0] Q2. (19 pts) Let B 0 1 8 and C = 6 3 0 0 0 1 5 4 1] determinants using the properties of determinants: Compute the following a) det($_35") b) det(3C) c) det(B25) d) det (((3C)B)')
3.8 Find the general solution of ſi 2-3 47 10 -1 2 2 X= 0 0...
3.8 Find the general solution of ſi 2-3 47 10 -1 2 2 X= 0 0 0 1 How many parameters do you have?
6. Consider the equation ſi 0 -17 |x=b, x>0. [1 1 1 Find all the basic...
6. Consider the equation ſi 0 -17 |x=b, x>0. [1 1 1 Find all the basic feasible solutions x for these values of b: []: [] [] 67 (-2): [ - ] [-] Draw a picture of the set of all b if x is feasible.
1 2 -1 0 0 1 0 0 -1 3 ſi 2 0 2 5 [10...
1 2 -1 0 0 1 0 0 -1 3 ſi 2 0 2 5 [10 (11 points) The matrix A= 2 1 3 2 7 reduces to R= 0 3 1 a 6 5 0 1 Let ui, , 13, 144, and us be the columns of U. (a) Determine, with justification, whether each of the following sets is linearly independent or linearly dependent. i. {u1, 12, 13) ii. {u1, 13, us} iii. {u2, 13} iv. {u1, 12, 13,...
A =10 ſi -2 -5 4 3 11 Jo 0 1 -2 0 -4 0 0...
A =10 ſi -2 -5 4 3 11 Jo 0 1 -2 0 -4 0 0 0 0 1 3 Lo 0 0 0 0 0 ] Describe all solutions of Ax = 0. x = x2 + 4
Find f: [0, 1] + R given by ſi if r = for any positive integer...
Find
f: [0, 1] + R given by ſi if r = for any positive integer n, JE) 10 otherwise, We were unable to transcribe this image
0 ſi 1 19. (5 points) Find the eigenvalues and eigenvectors of A= 0 2 2...
0 ſi 1 19. (5 points) Find the eigenvalues and eigenvectors of A= 0 2 2 Lo 03 1 0 20. (5 points) Show that A= 0 2 2 is diagonalizable by finding P and D such that p-1AP = D for [003] a diagonal D.
ſi 0 1 37 14 00 11 1. Compute the determinant of 10 4 11 5...
ſi 0 1 37 14 00 11 1. Compute the determinant of 10 4 11 5 using cofactors. Show your work. 12 0 1 2
(33 pts) This question is about the matrix = ſi 2 [3 2 0 4 1 6 3 1] 4 9 co (a) Find a lower triangular L and an upper triangular U so that A = LU. (b) Find the reduced row echelon form R = rref(A). How many independent columns in A? (c) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b
Find her Find ker (A) where A = where A-ſi 1 1] to . To write your answer properly, find vectors V1, ... , Vk so that ker(A) = span(V1, ..., Vk). 1 2 3 Let v1 = 1 , V2 = 2, V3 = 151. Determine if these vectors are 4 linearly independent. If they are not, then identify all redundant vectors.
T-1 2 5 ſi 0 0] Q2. (19 pts) Let B 0 1 8 and C = 6 3 0 0 0 1 5 4 1] determinants using the properties of determinants: Compute the following a) det($_35") b) det(3C) c) det(B25) d) det (((3C)B)')
3.8 Find the general solution of ſi 2-3 47 10 -1 2 2 X= 0 0 0 1 How many parameters do you have?
6. Consider the equation ſi 0 -17 |x=b, x>0. [1 1 1 Find all the basic feasible solutions x for these values of b: []: [] [] 67 (-2): [ - ] [-] Draw a picture of the set of all b if x is feasible.
1 2 -1 0 0 1 0 0 -1 3 ſi 2 0 2 5 [10 (11 points) The matrix A= 2 1 3 2 7 reduces to R= 0 3 1 a 6 5 0 1 Let ui, , 13, 144, and us be the columns of U. (a) Determine, with justification, whether each of the following sets is linearly independent or linearly dependent. i. {u1, 12, 13) ii. {u1, 13, us} iii. {u2, 13} iv. {u1, 12, 13,...
A =10 ſi -2 -5 4 3 11 Jo 0 1 -2 0 -4 0 0 0 0 1 3 Lo 0 0 0 0 0 ] Describe all solutions of Ax = 0. x = x2 + 4
Find
f: [0, 1] + R given by ſi if r = for any positive integer n, JE) 10 otherwise, We were unable to transcribe this image
0 ſi 1 19. (5 points) Find the eigenvalues and eigenvectors of A= 0 2 2 Lo 03 1 0 20. (5 points) Show that A= 0 2 2 is diagonalizable by finding P and D such that p-1AP = D for [003] a diagonal D.
ſi 0 1 37 14 00 11 1. Compute the determinant of 10 4 11 5 using cofactors. Show your work. 12 0 1 2
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