Sheet11MAT1163 Linear AlgebratextAssignment 2, 2014Due: 6 October 2014 12 noonQuestion 1Consider the matrix (a) Calculate the rref of the augmented matrix [A I] where I denotes the 3 by 3 identity matrix.(b) Is A invertible?1.00yesno(c) If your answer to (b) is yes, write down the inverse of A.(d) For each of the following statements decide if it is true or false:(i) A has two pivot positions.1.00TRUEFALSE(ii) The equation Ax = b has at least one solution1.00 for each bÎR3.(iii) The columns of A are linearly independent.1.00(iv) The linear transformation x®Ax is one to one.1.00Question 2The set of vectorsis not a subspace of R2.Prove this statement by(i) giving an example of a vextor x and a scalar α such that xÎS, but αxÏS.x =α =αx =(ii) giving an example of a vextor x and a vector y such that xÎS and yÎS, but x+yÏS.x =y =x + y =Question 3Consider the subspaces S1 and S2 defined by the equations and(a) The vector (1,3,1) belongs to one of the subspaces. Which one is it?1.00S1S2(b) Determine a basis for the subspace you found in (a). Use as many fields as you need.b1 =b2 =b3 =(c) Write down the augmented matrix of the system of equations that need to be solved to find the coordinates of (1,3,1) relative to the basis. (use as many columns as are needed) Write down the coordinates of (1,3,1) relative to the basis. (Use as many rows as you need)(d) Now consider the set of all vectors that belong to S1 and S2. Determine a vector equation that describes this set. (leave fields that are not needed free)x=s+t Give a geometric description of the set.The set is a 1.00lineplaneQuestion 4Consider the matrix (a) Determine the characteristic polynomial of A. (use the variable x in place of λ)6(b) Determine the eigenvalues of A(c) For each eigenvalue determine a basis for the correponding eigenspace.(d) Is the matrix A diagonalisable?1.00yesno(e) If your answer in (d) was yes write down a diagonal matrix D and an invertible matrix P so that PD=AP.D=P=Question 5Let u be a vector in R2, whose components satisfyPut and (a) IffindP=Q=(i) For calculatePx=Qx=(ii) For calculatePx=Qx=(iii) Find the eigenvalues and eigenvectors for P and QFor P:For Q:(b) IffindP=Q=(i) For calculatePx=Qx=(ii) For calculatePx=Qx=(iii) Find the eigenvalues and eigenvectors for P and QFor P:For Q:(c) How are the eigenvalues and eigenvectors of P and Q related? Complete the following sentences: If t is an eigenvalue for P, then the corresponding eigenvalue for Q is If x is an eigenvector for Q with eigenvalue t, then is an eigenvector for P eigenvalue text1 Sheet2 Sheet3 MAT1163 Linear Algebra Assignment 2 Due Date 6th October 2014, 12 noon Question 1 Consider the matrix               111 716 676 A . (a) Calculate the rref of the augmented matrix [A I] where I denotes the 33 identity matrix. (b) Is A.

1. Sheet11MAT1163 Linear AlgebratextAssignment 2, 2014Due: 6
October 2014 12 noonQuestion 1Consider the matrix (a)
Calculate the rref of the augmented matrix [A I] where I denotes
the 3 by 3 identity matrix.(b) Is A invertible?1.00yesno(c) If
your answer to (b) is yes, write down the inverse of A.(d) For
each of the following statements decide if it is true or false:(i)
A has two pivot positions.1.00TRUEFALSE(ii) The equation
Ax = b has at least one solution1.00 for each bÎR3.(iii)
The columns of A are linearly independent.1.00(iv) The linear
transformation x®Ax is one to one.1.00Question 2The set of
vectorsis not a subspace of R2.Prove this statement by(i)
giving an example of a vextor x and a scalar α such that xÎS, but
αxÏS.x =α =αx =(ii) giving an example of a vextor x and a
vector y such that xÎS and yÎS, but x+yÏS.x =y =x + y
=Question 3Consider the subspaces S1 and S2 defined by the
equations and(a) The vector (1,3,1) belongs to one of the
subspaces. Which one is it?1.00S1S2(b) Determine a basis for
the subspace you found in (a). Use as many fields as you
need.b1 =b2 =b3 =(c) Write down the augmented matrix of
the system of equations that need to be solved to find the
coordinates of (1,3,1) relative to the basis. (use as many
columns as are needed) Write down the coordinates of
(1,3,1) relative to the basis. (Use as many rows as you need)(d)
Now consider the set of all vectors that belong to S1 and S2.
Determine a vector equation that describes this set. (leave
fields that are not needed free)x=s+t Give a geometric
description of the set.The set is a 1.00lineplaneQuestion
4Consider the matrix (a) Determine the characteristic
polynomial of A. (use the variable x in place of λ)6(b)
Determine the eigenvalues of A(c) For each eigenvalue
determine a basis for the correponding eigenspace.(d) Is the
matrix A diagonalisable?1.00yesno(e) If your answer in (d)
was yes write down a diagonal matrix D and an invertible
matrix P so that PD=AP.D=P=Question 5Let u be a

2. vector in R2, whose components satisfyPut and (a)
IffindP=Q=(i) For calculatePx=Qx=(ii) For
calculatePx=Qx=(iii) Find the eigenvalues and eigenvectors
for P and QFor P:For Q:(b) IffindP=Q=(i) For
calculatePx=Qx=(ii) For calculatePx=Qx=(iii) Find the
eigenvalues and eigenvectors for P and QFor P:For Q:(c) How
are the eigenvalues and eigenvectors of P and Q related?
Complete the following sentences: If t is an eigenvalue for
P, then the corresponding eigenvalue for Q is If x is an
eigenvector for Q with eigenvalue t, then is an eigenvector for P
eigenvalue text1
Sheet2
Sheet3
MAT1163 Linear Algebra
Assignment 2
Due Date 6th October 2014, 12 noon
Question 1
Consider the matrix

3. 111
716
676
A .
(a) Calculate the rref of the augmented matrix [A I] where I
matrix.
(b) Is A invertible?
(c) If your answer to (b) is yes, write down the inverse of A.
(d) For each of the following statements decide if it is true or
false:
a. A has 2 pivot positions
b. The equation Ax=b has at least one solution for each b in R3.
c. The columns of A are linearly independent
one to one.
Question 2

4. of R
2.
yx,
Question 3
Consider the subspaces S1 and S2 of R
(a) The vector (1, 3, 1) belongs to one of the subspaces. Which
one is it?
(b) Determine a basis for the subspace you identified in (a)
(c) Write down the augmented matrix of the system of equations
that needs to be solved
to determine the coordinates of (1, 3, 1) relative to this basis.
Find the coordinates of
(1, 3, 1) relative to this basis.
(d) What is the dimension of this subspace?
(e) Now consider the set of all vectors in R3 which belong to
both S1 and S2. Determine a
vector equation that describes this set and give a geometric
description of it.

5. Question 4
Consider the matrix
201
810
265
A .

6. (a) Determine the characteristic polynomial of A.
(b) Determine the eigenvalues of A
(c) For each eigenvalue find a basis of the corresponding
eigenspace.
(d) Is the matrix A diagonalizable?
(e) If your answer to (d) is yes, write down a matrix P and a
matrix D such that AP=PD.
Question 5
Let u be a vector in R2 whose components satisfy 122
2
1
0
u ,
(i) find P and Q

7. (ii) calculate the image vectors Px and Qx for
0
1
1
0
x
(iii) Find the eigenvalues and corresponding eigenvectors of P
and Q .
(b) If

8. 22
22
u ,
(i) find P and Q
(ii) calculate the image vectors Px and Qx for
0
1

9. 1
0
x
(iii) Find the eigenvalues and corresponding eigenvectors of P
and Q .
(c) How are the eigenvalues and eigenvectors for P and Q
related?
then is an
eigenvalue for Q.
eigenvector for P with
eigenvalue …..