![4. (3 points) Let ſi 2 1] A= 0 4 3 [1 2 2 Compute the third column of A-1 by solving the equation Ax = es, where ez = 0 Hint:](http://img.homeworklib.com/questions/90f8e4b0-0858-11ec-b5ad-136848e1f043.png?x-oss-process=image/resize,w_560)
Homework Answers
![- (0-2)-(0-1941(0-on -1 (0-3)-0 (0-3)+1 (0-0) = -3 [about Istrow] and D2 = 2 ol 4 0 -1 (4-0)-2 (0-0)40 3 1= 4. [about tff row](http://img.homeworklib.com/questions/b5ab14f0-0858-11ec-ad9c-a96da828fe69.png?x-oss-process=image/resize,w_560)
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,...
1 -3 1 4 - 21 2. Let A= 1 1 3 06 0 1 -1...
1 -3 1 4 - 21 2. Let A= 1 1 3 06 0 1 -1 5-5 2 Do the columns of A span Rº? What does this mean for the general matrix equation Ax = b where be R*? 5 3. (a) Determine if A = 7 homogeneous equation. (8pts) -3 -4 5 2 1 has at least one free variable by solving the 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
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
1 1 1 0 -5 0 -77 0 2. 0 2 2. Let A be a...
1 1 1 0 -5 0 -77 0 2. 0 2 2. Let A be a 4 x 5 matrix whose reduced row echelon form is R 0 0 0 3 LO 0 0 0 0 For parts (b) and (c), write the solution in parametric vector form. (a) (2 points) Is the equation Ax b consistent for all b in R4? Why or why not? (b) (4 points) Solve the equation Ax = 0. 4 -3 -3 (c) (3...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A us...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...
ſi 4 01 Compute the inverse of the matrix A = 1 5 0 7 1...
ſi 4 01 Compute the inverse of the matrix A = 1 5 0 7 1 1
ſi 3 5] 1) (5 points) Compute the determinant of A= | 2 -2 1 using...
ſi 3 5] 1) (5 points) Compute the determinant of A= | 2 -2 1 using elemen- | 3 1 3 | tary row operations. No credit will be given for just the answer. Show enough work that I can see your elementary row operations used.
3. Let La A = 1 - 2 5 -3 2 5 0 -12-2 . L...
3. Let La A = 1 - 2 5 -3 2 5 0 -12-2 . L (a) (8 points) It turns out that the matrix equation Ax = b is consistent only for a special type of vector b where bi, b2, and b3 satisfy a certain equation. Find this equation. (b) (8 points) The set of all vectors satisfying the equation found in part (a) equals Span {W1, w2} Find wį and w2.
Question 3) (8 points) Consider the following matrix: A= ſi 4 0 0 28 3 12...
Question 3) (8 points) Consider the following matrix: A= ſi 4 0 0 28 3 12 2 11 -5 5 6 0 8 1 (a) Find a basis for the Rowspace(A). Then state the dimension of the Rowspace(A). (b) Find a basis for the Colspace(A). Then state the dimension of the Colspace(A). (e) Find a basis for the Nullspace(A). Then state the dimension of the Nullspace(A). (d) State and confirm the Rank-Nullity Theorem for this matrix.
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,...
1 -3 1 4 - 21 2. Let A= 1 1 3 06 0 1 -1 5-5 2 Do the columns of A span Rº? What does this mean for the general matrix equation Ax = b where be R*? 5 3. (a) Determine if A = 7 homogeneous equation. (8pts) -3 -4 5 2 1 has at least one free variable by solving the 0
(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
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
1 1 1 0 -5 0 -77 0 2. 0 2 2. Let A be a 4 x 5 matrix whose reduced row echelon form is R 0 0 0 3 LO 0 0 0 0 For parts (b) and (c), write the solution in parametric vector form. (a) (2 points) Is the equation Ax b consistent for all b in R4? Why or why not? (b) (4 points) Solve the equation Ax = 0. 4 -3 -3 (c) (3...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...
ſi 4 01 Compute the inverse of the matrix A = 1 5 0 7 1 1
ſi 3 5] 1) (5 points) Compute the determinant of A= | 2 -2 1 using elemen- | 3 1 3 | tary row operations. No credit will be given for just the answer. Show enough work that I can see your elementary row operations used.
3. Let La A = 1 - 2 5 -3 2 5 0 -12-2 . L (a) (8 points) It turns out that the matrix equation Ax = b is consistent only for a special type of vector b where bi, b2, and b3 satisfy a certain equation. Find this equation. (b) (8 points) The set of all vectors satisfying the equation found in part (a) equals Span {W1, w2} Find wį and w2.
Question 3) (8 points) Consider the following matrix: A= ſi 4 0 0 28 3 12 2 11 -5 5 6 0 8 1 (a) Find a basis for the Rowspace(A). Then state the dimension of the Rowspace(A). (b) Find a basis for the Colspace(A). Then state the dimension of the Colspace(A). (e) Find a basis for the Nullspace(A). Then state the dimension of the Nullspace(A). (d) State and confirm the Rank-Nullity Theorem for this matrix.
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