1. Calculate A 1000 for the matrix A defined in Holt, Section 6.4, problem 20, by the following process: Show that A can be diagonalized by finding an invertible matrix P and a diagonal matrix D such...

1 answer below »

1.
Calculate A1000
for the matrix A defined in Holt, Section 6.4, problem 20, by the following process:
Show that A can be diagonalized by finding an invertible matrix P and a diagonal matrix D such that A = PDP-1, and then calculate (PDP-1)1000.
2. Suppose that A is a diagonalizable matrix with distinct nonzero eigenvalues. Prove that A^2 has positive eigenvalues.
3. find the eigenvalues and a basis for each eigenspace for the given matrix:
A= [[1,-2][1,3]]
4. A vector space V and a subset S is given. Determine if S is a subspace of V, and if so, prove it. If not, give an example showing one of the conditions of definition 7.3 is not satisfied.
Def 7.3: a subset S of a vector space V is a subspace if S satisfied following 3 conditions

  1. S contains 0 the zero vector

  2. If u and v are in s, then u + v is also in s

  3. If c is a scalar and v is in s, then cv is also in s



V=C[-3,3] and S is the set of real valued functions f such that f(-1) + f(1) =0.
5. (same instructions as #4).
V=T(3,3) and S is the set of invertible linear transformations.
6. determine if the vector is in the subspace of R(2X3) given by SPAN {[[1,2,1][0,1,3]] , [[0,3,1][-1,1,0]]}
V= [[2,1,1][1,1,5]]
7. State whether following statement is true or false. Justify your answer.
if {v1,v2,v3} is a linearly independent set, then so is {v1, v2-v1, v3-v2+v1}
8. determine if the set
V
is a basis for V

V= {x^2-5x+3,3x^2-7x+5,x^2-x+1},
V=P^2
9. Find a basis for the subspace S and determine dim(s).
S is the subspace of P consisting of polynomials p such that p(0)=0.
10. Note:w Suppose that u1 and u2 are orthogonal vectors, with ||u1|| =3 and ||u2||=4. Find ||2u1-u2||.
11. Find a basis for S - for the subspace S.
S=SPAN [[1][1][-2]]


Document Preview:

1. Calculate A1000 for the matrix A defined in Holt, Section 6.4, problem 20, by the following process: Show that A can be diagonalized by finding an invertible matrix P and a diagonal matrix D such that A = PDP-1, and then calculate (PDP-1)1000. 2. Suppose that A is a diagonalizable matrix with distinct nonzero eigenvalues. Prove that A^2 has positive eigenvalues. 3. find the eigenvalues and a basis for each eigenspace for the given matrix: A= [[1,-2][1,3]] 4. A vector space V and a subset S is given. Determine if S is a subspace of V, and if so, prove it. If not, give an example showing one of the conditions of definition 7.3 is not satisfied. Def 7.3: a subset S of a vector space V is a subspace if S satisfied following 3 conditions S contains 0 the zero vector If u and v are in s, then u + v is also in s If c is a scalar and v is in s, then cv is also in s V=C[-3,3] and S is the set of real valued functions f such that f(-1) + f(1) =0. 5. (same instructions as #4). V=T(3,3) and S is the set of invertible linear transformations. 6. determine if the vector is in the subspace of R(2X3) given by SPAN {[[1,2,1][0,1,3]] , [[0,3,1][-1,1,0]]} V= [[2,1,1][1,1,5]] 7. State whether following statement is true or false. Justify your answer. if {v1,v2,v3} is a linearly independent set, then so is {v1, v2-v1, v3-v2+v1} 8. determine if the set V is a basis for V V= {x^2-5x+3,3x^2-7x+5,x^2-x+1}, V=P^2 9. Find a basis for the subspace S and determine dim(s). S is the subspace of P consisting of polynomials p such that p(0)=0. 10. Note:w Suppose that u1 and u2 are orthogonal vectors, with ||u1|| =3 and ||u2||=4. Find ||2u1-u2||. 11. Find a basis for S - for the subspace S. S=SPAN [[1][1][-2]]



Answered Same DayDec 20, 2021

Answer To: 1. Calculate A 1000 for the matrix A defined in Holt, Section 6.4, problem 20, by the following...

Robert answered on Dec 20 2021
126 Votes
Question 1
A= [[-2,2][0,0]]
Eigen values of A are
|A-λI|= 0
Hence (2+x)x=
So eigen values are x= 0, -2
Diagonal matrix D =[[ -2 0][0 0]]
P matrix are having corresponding eigen
vectors
P=[[1 , 1][1 0]]
1
1 1 2 0 1 1
A=PDP * *
0 1 0 0 0 1

      
      
     

1
1 1 2 0 1 1 2 2
A=PDP * *
0 1 0 0 0 1 0 0

         
        
       

Hence there exists matrix P and D such that above relation holds and D is a
diagonal matrix and P is an invertible matrix
2 1 1 2 1
1000 1000 1
A A*A PDP *PDP PD P
similarly
A PD P
  

  


Now D is a diagonal matrix
So
1000
1000 2 0D
0 0
 
  
 

1000
1000 12 0A P P
0 0

 
  
 

1000 1000 1000
1000
1 1 1 12 0 2 2
A * *
0 1 0 10 0 0 0
       
       
      

Question 2
A is a diagonizable matrix
There fore
1A=PDP
Hence there exists matrix P and D such that above relation holds and D is a
diagonal matrix and P is an invertible matrix
Now elements of the matrix D are Eigen values of matrix A
So elements in diagonal of D are a1,a2……an which are eigen values of A
Given a1, a2…..an are distinct and non zero
Eigen values of A^2 are the roots of characteristic equation
2A I
But
2 1 1 2 1A A*A PDP *PDP PD P    
Now if D is a diagonal matrix then D^2 is also a diagonal matrix
And Hence A^2 is a diagonalizable matrix.
Now eigen values of A^2 will the elements of the diagonal of D^2
Now elements of diagonal of D^2
Are a1^2 ,a2^2 ……an^2
None of the a1,a2…..an are zero
So there square will also not zero
If any ak is negative then its square is positive
Hence all the eigen values of A^2 are positive
Question 3
Eigen values of A are the roots of characteristic equation
A 0I 
2
1 2 1 0
0
1 3 0 1
4 5 0

 
   
...
SOLUTION.PDF

Answer To This Question Is Available To Download

Related Questions & Answers

More Questions »

Submit New Assignment

Copy and Paste Your Assignment Here