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Let C be an mxn matrix
And assume that a is an n-vector that is independent of the rows
of C. 
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Exercise 5. Let C be an mxn matrix. Assume that a is an n-vector that is...
A is mxn matrix Problem 7 (10pts) Prove any TWO of the following: Let A be...
A is mxn matrix
Problem 7 (10pts) Prove any TWO of the following: Let A be a mx n matrix. Then • (AA+)+ = AA+ and (A+A)+ = A+A • A+ = (ATA)+AT = AT (AAT)+ • A+ = (ATA)-IAT and A+A = In if rank(A) = n, • A+ = AT (AAT)-1 and AA+ = Im if rank(A) = m, • A+ = AT, if the columns of A are orthogonal, that is ATA=In
Show work 1. Assume you are given an mxn matrix H, an n-dimensional vector spacev h:VW...
Show work
1. Assume you are given an mxn matrix H, an n-dimensional vector spacev h:VW are there such that H RepBp(h)? B, and an m-dimensional vector space Wwith basis D. How many linear mappings A) None B) One C) It depends on m and n. D) Infinitely many 2. Which of the following matrices will change from the basi the basis(31),(7))? R2 to -1'1 3 -2 B) 3 2 A) C) D) 3. Assume you are given an n-dimensional...
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B...
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14]
QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
9. (2 pts per part) Let A be an m x n matrix, where m >...
9. (2 pts per part) Let A be an m x n matrix, where m > n, and suppose that the rank of A is n (i.e., A has full column rank). Briefly justify your answers to each question below. a. Which two of the following statements are true? i. There are no vectors in Nul(A). ii. There is no basis for Nul(A). iii. dim(Nul(A)) = 0 iv. dim(Nul(A)) = m – n b. Are the columns of A a...
In the next exercises, we consider square n X n matrices; I is the identity matrix...
In the next exercises, we consider square n X n matrices; I is the identity matrix (In MATLAB eye (n) gives a square n by n matrix). If ex is the column unit vector which components are all O's except the kth component which is equal to 1, i.e., 이".e1 = 101 0 then the identity matrix I is such that: 10 000 ei = [100 01T, , el 1000 11T. 0 0 T' 0 0 1 To generate in...
About linear algebra,matrix; 2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-en...
About linear algebra,matrix;
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-entry of the matrix A equals 0 if i + j is divisible by 5 and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u 9,64,-71,...
Let M be an n x n matrix with each entry equal to either 0 or 1. Let mij denote the entry in row i and column j. A diago...
Let M be an n x n matrix with each entry equal to either 0 or 1. Let mij denote the entry in row i and column j. A diagonal entry is one of the form mii for some i. Swapping rows i and j of the matrix M denotes the following action: we swap the values mik and mjk for k = 1,2, ... , n. Swapping two columns is defined analogously. We say that M is rearrangeable if...
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a) Show that A is symmetric (b) Show that N(A) S (c) Show tha...
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a) Show that A is symmetric (b) Show that N(A) S (c) Show that the rank of A must be 2.
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a)...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its rank. ii....
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its rank. ii. In case Z is just the n x 1 unit vector, i.e. Z- (1,....1)', what form does the vector Mz take? Note that x is any n- dimensional column vector
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its...
A is mxn matrix
Problem 7 (10pts) Prove any TWO of the following: Let A be a mx n matrix. Then • (AA+)+ = AA+ and (A+A)+ = A+A • A+ = (ATA)+AT = AT (AAT)+ • A+ = (ATA)-IAT and A+A = In if rank(A) = n, • A+ = AT (AAT)-1 and AA+ = Im if rank(A) = m, • A+ = AT, if the columns of A are orthogonal, that is ATA=In
Show work
1. Assume you are given an mxn matrix H, an n-dimensional vector spacev h:VW are there such that H RepBp(h)? B, and an m-dimensional vector space Wwith basis D. How many linear mappings A) None B) One C) It depends on m and n. D) Infinitely many 2. Which of the following matrices will change from the basi the basis(31),(7))? R2 to -1'1 3 -2 B) 3 2 A) C) D) 3. Assume you are given an n-dimensional...
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14]
QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
9. (2 pts per part) Let A be an m x n matrix, where m > n, and suppose that the rank of A is n (i.e., A has full column rank). Briefly justify your answers to each question below. a. Which two of the following statements are true? i. There are no vectors in Nul(A). ii. There is no basis for Nul(A). iii. dim(Nul(A)) = 0 iv. dim(Nul(A)) = m – n b. Are the columns of A a...
In the next exercises, we consider square n X n matrices; I is the identity matrix (In MATLAB eye (n) gives a square n by n matrix). If ex is the column unit vector which components are all O's except the kth component which is equal to 1, i.e., 이".e1 = 101 0 then the identity matrix I is such that: 10 000 ei = [100 01T, , el 1000 11T. 0 0 T' 0 0 1 To generate in...
About linear algebra,matrix;
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-entry of the matrix A equals 0 if i + j is divisible by 5 and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u 9,64,-71,...
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a) Show that A is symmetric (b) Show that N(A) S (c) Show that the rank of A must be 2.
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a)...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its rank. ii. In case Z is just the n x 1 unit vector, i.e. Z- (1,....1)', what form does the vector Mz take? Note that x is any n- dimensional column vector
Consider the n × n matrix M = In-Z(Z,Z)-1Z', where Z is n × K. i. Show that M is idempotent and find its...
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