The linear map T : R 3 ? R 4 is given by the matrix M(T) = ? ??? XXXXXXXXXX 1 1 3 ? ??? with respect...

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The linear map T : R 3 ? R 4 is given by the matrix M(T) = ? ??? XXXXXXXXXX 1 1 3 ? ??? with respect to the standard bases in R 3 and in R 4 . Find bases of Ker T and Im T. (2 marks) Problem A.2. Let S denote the set of polynomials in R[z] that have degree at most 5 and root 3 2 . Is S a vector space over R? Justify your answer. In the case of affirmative answer compute the dimension of S. (2 marks) Problem A.3. (a) Show that C = {z = x + y v -1 : x, y ? R} is a vector space of dimension 2 over R. (b) Define a linear map T from the vector space C to the vector space Mat(2, 2; R) of 2 × 2 matrices by the rule T : x + y v -1 7?  x y -y x . Compute dim Im T. (c) Show that T(z1z2) = T(z1)T(z2), where z1z2 is a usual multiplication of two complex numbers z1 and z2, and T(z1)T(z2) is the matrix multiplication of the two corresponding matrices. (2 marks) Date: 2nd semester XXXXXXXXXXMATH 2320 — ASSIGNMENT 1 Problem A.4. Suppose linear maps T ? L(V, W) and S ? L(W, U) of finitedimensional vector spaces are given. Show that dim Ker(ST) = dim Ker S + dim Ker T. (1 mark) Problem A.5. Show that the set of invertible n × n matrices (that is, the ones that correspond to invertible linear maps from F n to F n ) is a group with respect to the matrix multiplication and with the identity matrix I (which has ones along the main diagonal and zeroes otherwise). (1 mark)
	
Document Preview: MATH2320: LINEAR ALGEBRA|ASSIGNMENT 1
Name: Student No:
Due: Friday 30th August 2011 @ 10 am (Week 5)
You are required to complete this sheet and attach it to your worked solutions.
I expect all students to submit their own work as I am trying to assess your un-
derstanding of the basic ingredients of the course and your ability to do standard
mathematical problems.
Problems A.1, A.2 and A.3 are worth 2 marks (each), while Problems A.4 and A.5
are worth 1 mark. The entire assignment will give a mark out of 8.
3 4
Problem A.1. The linear map T :R !R is given by the matrix
0 1
2  1 0
B C
1 2 5
B C
M(T ) =
@ A
 1 1 1
1 1 3
3 4
with respect to the standard bases inR and inR . Find bases of KerT and ImT .
(2 marks)
Problem A.2. Let S denote the set of polynomials in R[z] that have degree at
3
most 5 and root . Is S a vector space overR? Justify your answer. In the case of
2
armative answer compute the dimension of S. (2 marks)
p
Problem A.3. (a) Show thatC =fz = x +y  1 : x;y2Rg is a vector space of
dimension 2 overR.
(b) Dene a linear mapT from the vector spaceC to the vector space Mat(2; 2;R)
of 2 2 matrices by the rule
 
p
x y
T : x +y  17! :
 y x
Compute dim ImT .
(c) Show that T (z z ) = T (z )T (z ), where z z is a usual multiplication of two
 XXXXXXXXXX
complex numbers z and z , and T (z )T (z ) is the matrix multiplication of the two
1 2 1 2
corresponding matrices. (2 marks)
Date: 2nd semester XXXXXXXXXXMATH2320|ASSIGNMENT 1
Problem A.4. Suppose linear maps T 2 L(V;W ) and S 2 L(W;U) of nite-
dimensional vector spaces are given. Show that
dim Ker(ST ) dim KerS + dim KerT:
(1 mark)
Problem A.5. Show that the set of invertible nn matrices (that is, the ones
n n
that correspond to invertible linear maps from F to F ) is a group with respect to
the matrix multiplication and with the identity matrix I (which has ones along the
main diagonal and zeroes otherwise). (1 mark)

Robert · Accepted Answer

Solutions 3 
(A) 
We need to show here the dimension of a vector space  1 : ,z x y x y      is 
2 over . 
Let 2:T  be a linear mapping from 2  onto , define by 
 , 1T x y x y    or  ,T x y x iy   
Let 1 2 1 2, , ,x x y y   
     
   
   
1 2 1 2 1 2 1 2
1 1 2 2
1 1 2 2
,
, ,
T x x y y x x i y y
x iy x iy
T x y T x y
     
   
 
 
Let  
   1 1 2 2
1 1 2 2
1 2 1 2
, ,T x y T x y
x iy x iy
x x and y y

  
  
 
Hence 2:T   is one to one. According to the theorem “If T is one to one linear 
mapping then its nullity or dimension of null space is zero.” 
i.e. ( ) 0.N T   
Since 2:T   is a linear mapping from 2  onto .

The linear map T : R 3 ? R 4 is given by the matrix M(T) = ? ??? XXXXXXXXXX 1 1 3 ? ??? with respect to the standard bases in R 3 and in R 4 . Find bases of Ker T and Im T. (2 marks) Problem A.2. Let...

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