Engineering Math

6.2 Exercises

1. Let
   {u1,⋅⋅⋅,un}
   be vectors in ℝⁿ. The parallelepiped determined by these vectors

   P (u1,⋅⋅⋅,un)

   is defined as

   { ∑n } P (u1,⋅⋅⋅,un) ≡ tkuk : tk ∈ [0,1] for all k . k=1

   Now let A be an n × n matrix. Show that

   {Ax : x ∈ P (u1,⋅⋅⋅,un )}

   is also a parallelepiped.

2. In the context of Problem 1, draw P
   (e1,e2)
   where e₁,e₂ are the standard basis vectors for ℝ². Thus e₁ =
   (1,0)
   ,e₂ =
   (0,1)
   . Now suppose

   ( ) 1 1 E = 0 1

   where E is the elementary matrix which takes the third row and adds to the first. Draw

   {Ex : x ∈ P (e1,e2)}.

   In other words, draw the result of doing E to the vectors in P

   (e1,e2)
   . Next draw the results of doing the other elementary matrices to P
   (e1,e2)
   . An elementary matrix is one which is obtained from doing one of the row operations to the identity matrix.

3. Determine which matrices are in row reduced echelon form.

   (a)

   ( ) 1 2 0 0 1 7

   (b)

   ( ) 1 0 0 0 |( 0 0 1 2 |) 0 0 0 0

   (c)

   ( ) 1 1 0 0 0 5 |( 0 0 1 2 0 4 |) 0 0 0 0 1 3

4. Row reduce the following matrices to obtain the row reduced echelon form. List the pivot columns in the original matrix.

   (a)

   ( ) | 1 2 0 3 | ( 2 1 2 2 ) 1 1 0 3

   (b)

   ( ) | 1 2 3 | || 2 1 − 2 || ( 3 0 0 ) 3 2 1

   (c)

   ( ) 1 2 1 3 |( − 3 2 1 0 |) 3 2 1 1

5. Find the rank of the following matrices. If the rank is r, identify r columns in the original matrix  which have the property that every other column may be written as a linear combination of these. Also find a basis for column space of the matrices.

   (a)

   ( 1 2 0 ) | | || 3 2 1 || ( 2 1 0 ) 0 2 1

   (b)

   ( ) 1 0 0 || 4 1 1 || |( 2 1 0 |) 0 2 0

   (c)

   ( ) 0 1 0 2 1 2 2 || 0 3 2 12 1 6 8 || |( 0 1 1 5 0 2 3 |) 0 2 1 7 0 3 4

   (d)

   ( 0 1 0 2 0 1 0 ) | | || 0 3 2 6 0 5 4 || ( 0 1 1 2 0 2 2 ) 0 2 1 4 0 3 2

   (e)

   ( 0 1 0 2 1 1 2 ) | 0 3 2 6 1 5 1 | || || ( 0 1 1 2 0 2 1 ) 0 2 1 4 0 3 1

6. Suppose A is an m × n matrix. Explain why the rank of A is always no larger than min
   (m, n)
   .
7. A matrix A is called a projection if A² = A. Here is a matrix.

   ( ) 2 0 2 |( 1 1 2 |) − 1 0 − 1

   Show that this is a projection. Show that a vector in the column space of a projection matrix is left unchanged by multiplication by A.

8. Let H denote span
   (( ) ( ) ( )) 1 2 1 2 , 4 , 3
   . Find the dimension of H and determine a basis.
9. Let H denote span
   ( ( ) ( ) ( ) ( )) 1 2 1 0 |( |( 2 |) ,|( 4 |) ,|( 3 |) ,|( 1 |)|) 0 0 1 1
   . Find the dimension of H and determine a basis.
10. Let H denote span
    ( ( ) ( ) ( ) ( )) 1 1 1 0 |( |( 2 |) ,|( 4 |) ,|( 3 |) ,|( 1 |)|) 0 0 1 1
    . Find the dimension of H and determine a basis.
11. Let M =
    {u = (u1,u2,u3,u4) ∈ ℝ4 : u3 = u1 = 0}
    . Is M a subspace? Explain.
12. Let M =
    { } u = (u1,u2,u3,u4) ∈ ℝ4 : u3 ≥ u1
    . Is M a subspace? Explain.
13. Let w ∈ ℝ⁴ and let M =
    { 4 } u = (u1,u2,u3,u4) ∈ ℝ : w ⋅u = 0
    . Is M a subspace? Explain.
14. Let M =
    {u = (u1,u2,u3,u4) ∈ ℝ4 : ui ≥ 0 for each i = 1,2,3,4}
    . Is M a subspace? Explain.
15. Let w,w₁ be given vectors in ℝ⁴ and define

    { 4 } M = u = (u1,u2,u3,u4) ∈ ℝ : w ⋅u = 0 and w1 ⋅u = 0 .

    Is M a subspace? Explain.

16. Let M =
    {u = (u1,u2,u3,u4) ∈ ℝ4 : |u1| ≤ 4}
    . Is M a subspace? Explain.
17. Let M =
    { } u = (u1,u2,u3,u4) ∈ ℝ4 : sin (u1) = 1
    . Is M a subspace? Explain.
18. Suppose
    {x1,⋅⋅⋅,xk}
    is a set of vectors from Fⁿ. Show that span
    (x1,⋅⋅⋅,xk)
    contains 0.
19. Prove the following theorem: If A,B are n×n matrices and if AB = I, then BA = I and B = A⁻¹. Hint: First note that if AB = I, then it must be the case that A is onto. Explain why this requires span
    (columns of A)
    = Fⁿ. Now explain why, this requires A to be one to one. Next explain why A
    (BA − I)
    = 0 and why the fact that A is one to one implies BA = I.
20. Here are three vectors. Determine whether they are linearly independent or linearly dependent.

    ( )T ( )T ( )T 1 2 0 , 2 0 1 , 3 0 0

21. Here are three vectors. Determine whether they are linearly independent or linearly dependent.

    ( )T ( )T ( )T 4 2 0 , 2 2 1 , 0 2 2

22. Here are three vectors. Determine whether they are linearly independent or linearly dependent.

    ( )T ( )T ( )T 1 2 3 , 4 5 1 , 3 1 0

23. Here are four vectors. Determine whether they span ℝ³. Are these vectors linearly independent?

    ( ) ( ) ( ) ( ) 1 2 3 T , 4 3 3 T , 3 1 0 T , 2 4 6 T

24. Here are four vectors. Determine whether they span ℝ³. Are these vectors linearly independent?

    ( ) ( ) ( ) ( ) 1 2 3 T , 4 3 3 T , 3 2 0 T , 2 4 6 T

25. Determine whether the following vectors are a basis for ℝ³. If they are, explain why they are and if they are not, give a reason and tell whether they span ℝ³.

    ( )T ( )T ( )T ( )T 1 0 3 , 4 3 3 , 1 2 0 , 2 4 0

26. Determine whether the following vectors are a basis for ℝ³. If they are, explain why they are and if they are not, give a reason and tell whether they span ℝ³.

    ( )T ( )T ( )T 1 0 3 , 0 1 0 , 1 2 0

27. Determine whether the following vectors are a basis for ℝ³. If they are, explain why they are and if they are not, give a reason and tell whether they span ℝ³.

    ( ) ( ) ( ) ( ) 1 0 3 T , 0 1 0 T , 1 2 0 T , 0 0 0 T

28. Determine whether the following vectors are a basis for ℝ³. If they are, explain why they are and if they are not, give a reason and tell whether they span ℝ³.

    ( ) ( ) ( ) ( ) 1 0 3 T , 0 1 0 T , 1 1 3 T , 0 0 0 T

29. Consider the vectors of the form

    ( ( ) ) |{ 2t+ 3s |} |( s− t |) : s,t ∈ ℝ . |( t+ s |)

    Is this set of vectors a subspace of ℝ³? If so, explain why, give a basis for the subspace and find its dimension.

30. Consider the vectors of the form

    (( ) ) |||{| 2t+ 3s+ u | |||} || s− t || : s,t,u ∈ ℝ . |||( t+ s ) ||| ( u )

    Is this set of vectors a subspace of ℝ⁴? If so, explain why, give a basis for the subspace and find its dimension.

31. Consider the vectors of the form

    (|( 2t+ u ) )| ||{| t+ 3u | ||} || || : s,t,u,v ∈ ℝ . |||(( t+ s+ v ) |||) u

    Is this set of vectors a subspace of ℝ⁴? If so, explain why, give a basis for the subspace and find its dimension.

32. If you have 5 vectors in F⁵ and the vectors are linearly independent, can it always be concluded they span F⁵? Explain.
33. If you have 6 vectors in F⁵, is it possible they are linearly independent? Explain.
34. Suppose A is an m × n matrix and
    {w1,⋅⋅⋅,wk}
    is a linearly independent set of vectors in A
    n (F )
    ⊆ F^m. Now suppose A
    (zi)
    = w_i. Show
    {z1,⋅⋅⋅,zk}
    is also independent.
35. Suppose V,W are subspaces of Fⁿ. Show V ∩W defined to be all vectors which are in both V and W is a subspace also.
36. Suppose V  and W both have dimension equal to 7 and they are subspaces of F¹⁰. What are the possibilities for the dimension of V ∩ W? Hint: Remember that a linear independent set can be extended to form a basis.
37. Suppose V  has dimension p and W has dimension q and they are each contained in a subspace, U which has dimension equal to n where n > max
    (p,q)
    . What are the possibilities for the dimension of V ∩ W? Hint: Remember that a linear independent set can be extended to form a basis.
38. If b≠0, can the solution set of Ax = b be a plane through the origin? Explain.
39. Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.
40. Suppose a system of linear equations has a 2 × 4 augmented matrix and the last column is a pivot column. Could the system of linear equations be consistent? Explain.
41. Suppose the coefficient matrix of a system of n equations with n variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.
42. Suppose there is a unique solution to a system of linear equations. What must be true of the pivot columns in the augmented matrix.
43. State whether each of the following sets of data are possible for the matrix equation Ax = b. If possible, describe the solution set. That is, tell whether there exists a unique solution no solution or infinitely many solutions.
    1. A is a 5 × 6 matrix, rank
       (A)
       = 4 and rank
       (A|b)
       = 4. Hint: This says b is in the span of four of the columns. Thus the columns are not independent.
    2. A is a 3 × 4 matrix, rank
       (A)
       = 3 and rank
       (A |b )
       = 2.
    3. A is a 4 × 2 matrix, rank
       (A)
       = 4 and rank
       (A|b)
       = 4. Hint: This says b is in the span of the columns and the columns must be independent.
    4. A is a 5 × 5 matrix, rank
       (A)
       = 4 and rank
       (A|b)
       = 5. Hint: This says b is not in the span of the columns.
    5. A is a 4 × 2 matrix, rank
       (A)
       = 2 and rank
       (A |b )
       = 2.
44. Suppose A is an m×n matrix in which m ≤ n. Suppose also that the rank of A equals m. Show that A maps Fⁿ onto F^m. Hint: The vectors e₁,
    ⋅⋅⋅
    ,e_m occur as columns in the row reduced echelon form for A.
45. Suppose A is an m×n matrix in which m ≥ n. Suppose also that the rank of A equals n. Show that A is one to one. Hint: If not, there exists a vector x such that Ax = 0, and this implies at least one column of A is a linear combination of the others. Show this would require the column rank to be less than n.
46. Explain why an n × n matrix A is both one to one and onto if and only if its rank is n.
47. Suppose A is an m × n matrix and B is an n × p matrix. Show that

    dim (ker(AB )) ≤ dim (ker(A))+ dim(ker(B)).

    Hint: Consider the subspace, B

    (Fp)
    ∩ ker
    (A )
    and suppose a basis for this subspace is

    {w1,⋅⋅⋅,wk }.

    Now suppose

    {u1,⋅⋅⋅,ur}
    is a basis for ker
    (B)
    . Let
    {z1,⋅⋅⋅,zk}
    be such that Bz_i = w_i and argue that

    ker(AB ) ⊆ span(u1,⋅⋅⋅,ur,z1,⋅⋅⋅,zk).

    Here is how you do this. Suppose ABx = 0. Then Bx ∈ ker

    (A)
    ∩ B
    p (F )
    and so Bx = ∑ _i=1^kBz_i showing that

    ∑k x − zi ∈ ker(B ). i=1

48. Explain why Ax = 0 always has a solution even when A⁻¹ does not exist.
    1. What can you conclude about A if the solution is unique?
    2. What can you conclude about A if the solution is not unique?
49. Let A be an n×n matrix and let x be a nonzero vector such that Ax = λx for some scalar λ. When this occurs, the vector x is called an eigenvector and the scalar λ is called an eigenvalue. It turns out that not every number is an eigenvalue. Only certain ones are. Why? Hint: Show that if Ax = λx, then
    (A − λI)
    x = 0. Explain why this shows that
    (A − λI)
    is not one to one and not onto.
50. Let A be an n × n matrix and consider the matrices
    { 2 n2} I,A, A ,⋅⋅⋅,A
    . Explain why there exist scalars, c_i not all zero such that

    n2 ∑ ciAi = 0. i=1

    Then argue there exists a polynomial, p

    (λ)
    of the form

    m m −1 λ + dm−1λ + ⋅⋅⋅+ d1λ + d0

    such that p

    (A )
    = 0 and if q
    (λ)
    is another polynomial such that q
    (A)
    = 0, then q
    (λ)
    is of the form p
    (λ)
    l
    (λ)
    for some polynomial, l
    (λ)
    . This extra special polynomial, p
    (λ)
    is called the minimal polynomial. Hint: You might consider an n×n matrix as a vector in Fⁿ² . What would be a basis for this set of matrices?