![hapter 1 Systems of Linear Equations and Matrices *21. Suppose that A is n × m and B is m × n so that AB is n × n. Show that AB is no invertible if n> m. [Hint: Show that there is a nonzero vector x such that AB then apply Theorem 6.] and 22.) Use the methods of this section to find the inverses of the following matrices complex entries: 1- 0 a. sin θ cos θ 0 23. Show that for every real number θ the matrix | cos θ -sin θ 0 | i invertible and find its inverse. 24) Calculate the inverse of A = | 0 3 0 0 0 4](http://img.homeworklib.com/questions/dc3947a0-a431-11eb-909c-6703329bacf0.png?x-oss-process=image/resize,w_560)
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Please answer # 22 and 24
hapter 1 Systems of Linear Equations and Matrices *21. Suppose...
Differention Equations - Can someone answer the checked numbers please? Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve fo...
Differention Equations - Can someone answer the checked
numbers please?
Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...
2. Consider the following set of complex 2 x 2 matrices where i = -1: H...
2. Consider the following set of complex 2 x 2 matrices where i = -1: H = a + bi -c+dil Ic+dia-bi Put B = {1, i, j, k} where = = {[ctdie met di]|1,3,c,dex} 1-[ ), : = [=]. ; = [i -:], « =(: :] . (a) Show that H is a subspace of the real vector space of 2 x 2 matrices with entries from C, that is, show H is closed under matrix addition and multi-...
Could someone pls explain question 9 (e)? 9. Consider the set of matrices F = a)...
Could someone pls explain question 9 (e)?
9. Consider the set of matrices F = a) Show that AB BA for all A, B E F b) Show that every A E F\ {0} is invertible and compute A-. c) Show that F is a field d) Show that F can be identified with C e) What form of matrix in F corresponds to the modđulus-argument form of a complex number Comment on the geometric significance. Solution a) Let A...
21 please inteb CORE 17 20. The matrices in the last two Exercises were the standard...
21 please
inteb CORE 17 20. The matrices in the last two Exercises were the standard matrices of the operators [proji] and refli], respectively, where L is a line through the origin in R2 with unit direction vector (a, b) See Exercise 25 in Section 2.2. Give a geometric argument as to why one of these matrices is invertible and the other matrix is not invertible. Explain also the geometric significance of the inverse of the invertible matrix. For Exercises...
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that...
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that...
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define...
2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define the spin matrices, s.-슬&z, Šv = , S.-출.. We'll use the eigenkets of S that, for the spin half system, they can be represented by the spinors, a) Show, by matrix multiplication that |+) and |-) are eigenstates of the S operator and determine the eigenvalues. Show that they are not eigenstates of S and Sy b) Show that the matrix squares s ,...
LINEAR ALGEBRA: PLEASE FOLLOW THE COMMENT and please tell me what is the rotate matrix and...
LINEAR ALGEBRA: PLEASE FOLLOW THE COMMENT and please
tell me what is the rotate matrix and why there is cos@ and -sin@ i
think it should be cos@ and sin@ on the first row
For each of the following linear operators on R2,
find the matrix representation of the transformation
with respect to the homogeneous coordinate
system:
(a) The transformation L that rotates each vector
by 120◦ in the counterclockwise direction
(b) The transformation L that translates each point
3...
L. Answer True or False. Justify your answer (a) Every linear system consisting of 2 equations...
L. Answer True or False. Justify your answer (a) Every linear system consisting of 2 equations in 3 unknowns has infinitely many solutions (b) If A. B are n × n nonsingular matrices and AB BA, then (e) If A is an n x n matrix, with ( +A) I-A, then A O (d) If A, B two 2 x 2 symmetric matrices, then AB is also symmetric. (e) If A. B are any square matrices, then (A+ B)(A-B)-A2-B2 2....
Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0)...
Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0) = sin() cos(0) Consider now the unit vector v = [1,0)" in a two dimensional plane. Compute R(O)v. Repeat your computations this time using w = [0, 1]". What do you observe? Try thinking in terms of pictures, look at the pair of vectors before and after the action of R(O). 23. You may have recognised the two vectors in the previous question to...
Differention Equations - Can someone answer the checked
numbers please?
Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...
2. Consider the following set of complex 2 x 2 matrices where i = -1: H = a + bi -c+dil Ic+dia-bi Put B = {1, i, j, k} where = = {[ctdie met di]|1,3,c,dex} 1-[ ), : = [=]. ; = [i -:], « =(: :] . (a) Show that H is a subspace of the real vector space of 2 x 2 matrices with entries from C, that is, show H is closed under matrix addition and multi-...
Could someone pls explain question 9 (e)?
9. Consider the set of matrices F = a) Show that AB BA for all A, B E F b) Show that every A E F\ {0} is invertible and compute A-. c) Show that F is a field d) Show that F can be identified with C e) What form of matrix in F corresponds to the modđulus-argument form of a complex number Comment on the geometric significance. Solution a) Let A...
21 please
inteb CORE 17 20. The matrices in the last two Exercises were the standard matrices of the operators [proji] and refli], respectively, where L is a line through the origin in R2 with unit direction vector (a, b) See Exercise 25 in Section 2.2. Give a geometric argument as to why one of these matrices is invertible and the other matrix is not invertible. Explain also the geometric significance of the inverse of the invertible matrix. For Exercises...
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define the spin matrices, s.-슬&z, Šv = , S.-출.. We'll use the eigenkets of S that, for the spin half system, they can be represented by the spinors, a) Show, by matrix multiplication that |+) and |-) are eigenstates of the S operator and determine the eigenvalues. Show that they are not eigenstates of S and Sy b) Show that the matrix squares s ,...
LINEAR ALGEBRA: PLEASE FOLLOW THE COMMENT and please
tell me what is the rotate matrix and why there is cos@ and -sin@ i
think it should be cos@ and sin@ on the first row
For each of the following linear operators on R2,
find the matrix representation of the transformation
with respect to the homogeneous coordinate
system:
(a) The transformation L that rotates each vector
by 120◦ in the counterclockwise direction
(b) The transformation L that translates each point
3...
L. Answer True or False. Justify your answer (a) Every linear system consisting of 2 equations in 3 unknowns has infinitely many solutions (b) If A. B are n × n nonsingular matrices and AB BA, then (e) If A is an n x n matrix, with ( +A) I-A, then A O (d) If A, B two 2 x 2 symmetric matrices, then AB is also symmetric. (e) If A. B are any square matrices, then (A+ B)(A-B)-A2-B2 2....
Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0) = sin() cos(0) Consider now the unit vector v = [1,0)" in a two dimensional plane. Compute R(O)v. Repeat your computations this time using w = [0, 1]". What do you observe? Try thinking in terms of pictures, look at the pair of vectors before and after the action of R(O). 23. You may have recognised the two vectors in the previous question to...
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