Unit IV – Vector Spaces

Part-B

1. Suppose a 3x5 coefficient matrix for a system has three pivot columns. Is the system
consistent? Why or why not?
2. Define a vector space.
3. Define subspace.
4. Let V  R 3 , Check whether W is a subspace or not. Where W  (a, b, c) : a  0

1 2 3 4

5. Determine by inspection if the given set is linearly dependent: 7, 0, 1, 1
6 9 5 8

6. Determine whether the given vectors are linearly independent. Justify your answer.

1 1  1
1,   2 , 0
    
1  1   1 

7. Define Basis and Dimension of a vectorspace.

8. Define Null space.
9. Define Column space.
10. Define Row space.
 3 
 2 4  2 1  
    2
11. Let A    2  5 7 3  and u    ,
 3 1
 7  8 6   
 0 
 
Determine if u is in Nul A. Could u be in Col A?

 1  6  2    4
       
12. Let v1    3 , v 2   2 , v3    2  and v 4    8  .Find a basis for the subspace
 4    1  3   9 
       

spanned by
v1, v2 , v3 , v4 
13. If V  A  B then show that dimV=dimA+dimB

PART-C

1.Find the general solution the of the system by Row reduction algorithm
x1  2 x 2  x3  3x 4  0 ;  2 x1  4 x 2  5x3  5x 4  3 ; 3x1  6 x 2 6 x3  3x 4  2

2. Apply elementary row operations to transform the following matrix into reduced echelon
1 2 3 4 
form 4 5 6 7 and identify the pivot columns.
6 7 8 9

3. Apply elementary row operations to transform the following matrix into reduced echelon
 0 3 6 4 9 
1  2 1 3 1 
form  and identify the pivot columns.
 2  3 0 3  1
 
1 4 5  9  7

4.Is the set of all pairs of real numbers with the operations
x  y  x' y'  x  x'1, y  y'1 and k x, y  kx, ky a vector space? Verify all the
axioms and list the axioms that fail to hold if it is not a vector space.

5.Prove that the set S  M 1 , M 2 , M 3 , M 4 is a basis for the vector space M 2 x 2 of a 2x2

1 0 0 1 0 0  0 0 
matrices where M 1    ,M2    M3    ,M4   .
0 0 0 0 1 0 0 1 

6. Prove that for n  0 , the set Pn of polynomials of degree n consists of all polynomials of

the form P(t )  a0  a1 t       a n t n is a vector space.

7. Let V be a set of all real numbers with the operations x  y  xy & cx  x

c
Verify Vis
a vector space or not .

3   6 0 3

     
8. Let u  2 , v  1 , w   5 and z   7 
       
 4  7   2   5

(i) Are the sets u, v, u, w, u, zw, z, v, w, v, z each linearly independent?

(ii) Is u, v, w, z linearly dependent?

(iii)To determine if u, v, w, z is linearly dependent , is it wise to check if say w is a linear

combination of u, v,& z.

9.Find the basis and dimension of the subspace spanned by

 1  2 1  1   2
2  4 3  4 7 
         
v1   1 , v 2   2 , v 3  2 , v 4  5 , v5  3
         
3 6  2 1   3
 4   8  6 8  9 

  9 7   4  9
7   4 5   
10. Show that w is in the subspace of R4 spanned by v1    , v 2    v3    , w   4 
4  2   1  4
       
8  9   7   7 

 2 1 1  6 8 
 
 1 2 4 3 2
11.For the matrix A 
  7 8 10 3  10 
 
 4 5 7 0 4 


i) Find the bases for Nul A, Col A, Row A

ii) Find rank A and dim(Nul A)

Unit V – Orthogonality and Inner Product Spaces

Part – B
 0    4
   
1. Calculate the distance between u    5  and v    1  .
 2   8 
   
2. State geometric interpretation of the orthogonal projection of y onto u .

7  4
3. Let y    , u    , find the orthogonal projections of y onto u .
 6  2
3  1   1 / 2
4. Verify that u1 , u 2 , u3 is an orthogonal set, where u1  1, u 2   2 , u3    2 
   
1  1   7 / 2 

5. Define an orthogonal basis.

6. Define and orthonormal basis.
7. verify that u1 ,u2  is an orthonormal set, where

u1  1/ 3, 2 / 3, 2 / 3, u2  (2 / 3, 1/ 3,  2 / 3)
 PART-C
1. Show that u1, u2 , u3  is an orthogonal basis for R3 . Also express x as a linear combination

1  1 2 8

     
of u ’s where u1  0 , u2  4 , u3  1 and x   4
       
1  1   2   3

 2 7 
2. Let y   , u    compute the distance from y to the line through the origin and u.
6  1 
1  4
3. Compute the orthogonal projection of   onto the line through   and the origin.
1  2 
0   4
4. Let u   5, z    1 (i) Find a unit vector in the direction of z. (ii) Find the distance
 
 2   8 

between u and Z.
5. Let W be the subspace spanned by the u’s, write y as the sum of a vector in W and a vector
1 1  5 
   
orthogonal to W , where y  3 , u1  3 , u 2  1
     
5  2 4

5. Let
u1 , u 2 , u3 , u 4 be an orthogonal basis for R4. Write x as the sum of vectors one in

and the other in spanu4 where

spanu1 , u2 , u3 

0 3 1 5  10 

1 5 0  3  8
u1   , u2   , u3    , u4   , x   
  4 1 1   1 2
         
  1 1   4 1 0

 
6. Verify that u1 ,u2 is an orthogonal set and then find the orthogonal projection y onto
 1 1  1
spanu1 ,u2 where y  4 , u1  1 , u2   1 
   
     
 3  0  0 

7. Let
 3 1 1 0
 4 1 0   1
y   , u1   , u 2    , u 3   
5  0 1 1
       
6   1 1  1
Write y as a sum of two vectors one in
Spanu1 , u 2 , u3 and another orthogonal to spanu1 , u 2 , u3 .

8. Find the closest point to y in the subspace w spanned by u1 and u2 where

3 3 1
1 1  1
y   , v1   , v 2   
5  1 1
     
1 1  1

9 Let R4 have the Euclidean inner product. Use Gram Schmidt orthogonalization process
transform the basis u1, u2 , u3 , u4  into orthonormal basis where

u1  0,2,1,0, u2  1,1,0,0, u3  1,2,0,1, u4  1,0,0,1

10. Use Gram Schmidt process to produce an orthonormal basis for w whose basis has the
1  7 
  4   7 
   
vectors  0  ,  4
   
1  1 

11. Consider the vectorspaceP(t) with innerproduct

f , g   f (t ) g (t )dt . Apply Gram Schmidt algorithm to 1, t , t 2 to obtain orthogonal

1

set
 f1 , f 2 , f3  with integral coefficient.

   
12. Let u  u1 ,u 2 , v  v1 ,v 2 be two vectors in R . Verify that the Euclidean inner product
2

u, v  3u1v1  2u2 v2 satisfies the following innerproduct axioms i) u, v   v, u , ii)
u  v, w  u, w  v, w
1

13.If f(t)= t+2, g(t)=t -2t-3 find the inner product f , g defined by f , g   f (t ) g (t )dt .
2
0