Download Exercise Set iv 1. Let W1 be a set of all vectors (a, b, c, d) in R4 such

Exercise Set iv
1. Let W1 be a set of all vectors (a, b, c, d) in R4 such that a + d = 0 and W2 be a set of all vectors
(a, b, c, d) in R4 such that ad = 0. Is W1 a subspace of R4 ? Is W2 a subspace of R4 ?
2. Let u~1 = (1, 1, 0), u~2 = (1, 0, 1), u~3 = (0, 1, 1)
(a) Show that {u~1 , u~2 , u~3 } is linearly independent.
(b) Express ~v = (1, 2, 3) as a linear combination of u~1 , u~2 , u~3 .
3. (a) Give an example of a system of three vectors {u~1 , u~2 , u~3 } in R3 with the following properties:
i. {u~1 , u~2 , u~3 } is linearly dependent.
ii. Each of the three systems {u~1 , u~2 }, {u~1 , u~3 }, {u~2 , u~3 } is linearly independent.
Give reasons for your answer and the geometric interpretation.
(b) Let A and B are arbitrary 2 × 2 matrices. Prove that the set W of 2 × 2 matrices X such that
AX = X T B is a subspace of the space V of all real 2 × 2 matrices. (Hint: (X + Y )T + X T + Y T .)
(c) Find a basis for the space W from part (b), when
1 2
2
A=
, B=
3 1
3
3
1
4. (a) For which real value of λ do the following vectors
v~1 = (λ, 1, 1), v~2 = (1, λ, 1, ), v~3 = (0, 3, 1)
form a linearly dependent set in R3 ?
(b) Prove that the set W of all 2 × 2 matrices of the form
a
a−b
, a, b ∈ R
a + 2b
b
is a subspace of the vector space V of all 2 × 2 matrices.
5. Let W be the set of all vectors in R4 such that their first coordinate is twice their third coordinate and
that the sum of their second and fourth coordinates is zero.
(a) Prove that W is subspace in R4 .
(b) Write down a homogeneous system of linear equations for which W is the set of solutions.
(c) Represent W as a span of a system of vectors.
6. Let V = R3 . Let ~v1 = (1, 1, 0), ~v2 = (1, 0, 1), ~v3 = (0, 1, 0), ~v4 = (4, −1, 6) and ~v5 = (0, 0, 0) be vectors
in V . Determine whether each of the following sets form a basis for V . Explain.
(a) S1 = {~v1 , ~v2 , ~v3 , ~v4 }.
(b) S2 = {~v1 , ~v2 , ~v5 }.
(c) S3 = {~v1 , ~v2 , ~v3 }.
7. Let

1 0
A= 8 4
−2 1

−1
2
2
0 .
1 −4
(a) Find a basis for the row space of A.
(b) What is rank(A), nullity(A)?
8. Given the vector space V and the subset W ⊂ V . Determine whether W is a subspace of V in the
following cases. Explain.
(a) V = R3 , W = {(x, y, z) ∈ R3 : (x, y − 2, z) · (1, 4, 2) = 0}.
a b
(b) V = M2×2 , W =
:a+b+c=0 .
c d
Note: M2×2 is the set of all 2 × 2 matrices with usual matrix addition and scalar multiplication by
reals.
9. Given
A=
1
1
1
0
, B=
1
0
0
1
0
0
, C=
1
0
, D=
0
1
0
0
.
(a) Show that S = {A, B, C, D} is a basis for M2×2 .
(b) Represent the matrix
E=
2
3
0
1
as a linear combination of the matrices from S.
10. Suppose that S = {~v1 , ~v2 } is a linearly independent set and that the vector ~v3 does not belong to
span(S). Prove that S 0 = {~v1 , ~v2 , ~v3 } is also a linearly independent set.
11. Find bases for the row space of A, column space of A. Find rank(A) and nullity(A).


1
2 3 4
A =  −1 −1 2 2  .
1
1 4 4
12. In each part, write down a matrix with required property or explain why no such matrix exit.

 

0
3
(a) Column space contains  0  ,  0 , row space contains 1 1 , 1 2 .
2
0
(b) Column space = R2 , row space = R4 .
 


1
3
(c) Column space contains  3  ,  0 , rank=3.
1
3


 
5
2
(d) Column space has basis  5 , null space has basis  2 .
5
1
13. In each part a set of objects is given together with operations of addition (⊕) and scalar multiplication
(). Determine which set(s) are vector spaces under the given operations. Justify your answer.
(a) The set of all triples of real numbers (x, y, z) with the operations
(x, y, z) ⊕ (x0 , y 0 , z 0 ) = (x + x0 , y + y 0 , z + z 0 )
and k (x, y, z) = (kx, y, z).
(b) The set of all pairs of real numbers of the form (1, x) with the operations
(1, y) ⊕ (1, y 0 ) = (1, y + y 0 )
and k (1, y) = (1, ky).
14. Determine which of the following are subspaces of M22 .
(a) all 2 × 2 matrices with integer entries.
(b) all matrices
a
c
b
d
a b
0 c
where a + b + c + d = 0.
(c) all 2 × 2 matrices A such that |A| = 0.
(d) all matrices of the form
15. Determine which of the following are subspaces of Mnn .
(a) all n × n matrices A such that Tr(A) = 0.
(b) all n × n matrices A such that AT = −A.
2
.
(c) all n × n matrices A such that |A| =
6 0.
(d) all n × n matrices A such that AB = BA for a fixed n × n matrix B.
16. In each part determine whether the given vectors span R3 .
(a) ~v1 = (3, 1, 4), ~v2 = (2, −3, 5), ~v3 = (5, −2, 9), ~v4 = (1, 4, −1).
(b) ~v1 = (1, 2, 6), ~v2 = (3, 4, 1), ~v3 = (4, 3, 1), ~v4 = (3, 3, 1).
17. (a) Show that the vectors ~v1 = (0, 3, 1, −1), ~v2 = (6, 0, 5, 1), ~v3 = (4, −7, 1, 3) form a linearly dependent set in R4 .
(b) Express each vector as a linear combination of the other two.
18. Prove: For any vectors ~u, ~v and w,
~ the vectors ~u − ~v , ~v − w
~ and w
~ − ~u form a linearly dependent set.
19. Let ~u, ~v , w
~ be linearly independent. Prove that ~u + ~v , ~v + w,
~ w
~ + ~u are also linearly independent.
Compare this result with the previous problem. Why are the determinants
1 0 1 1
0 −1 −1
1
0 and 1 1 0 0 1 1 0 −1
1 relevant.
20. Which of the following sets of vectors in R4 are linearly dependent?
(a) (3, 8, 7, −3), (1, 5, 3, −1), (2, −1, 2, 6), (1, 4, 0, 3).
(b) (0, 0, 2, 2), (3, 3, 0, 0), (1, 1, 0, −1).
(c) (0, 3, −3, −6), (−2, 0, 0, −6), (0, −4, −2, −2), (0, −8, 4, −4).
(d) (3, 0, −3, 6), (0, 2, 3, 1), (0, −2, −2, 0), (−2, 1, 2, 1).
21. Which of the following sets of vectors are bases for R3 ?
(a) (1, 0, 0), (2, 2, 0), (3, 3, 3).
(b) (3, 1, −4), (2, 5, 6), (1, 4, 8).
(c) (2, −3, 1), (4, 1, 1), (0, −7, 1).
(d) (1, 6, 4), (2, 4, −1), (−1, 2, 5).
22. Determine the dimensions of the following subspaces of R4 .
(a) All vectors of the form (a, b, c, 0).
(b) All vectors of the form (a, b, c, d) where d = a + b and c = a − b.
(c) All vectors of the form (a, b, c, d) where a = b = c = d.
23. Find a basis for the null space, row space and column space of A.


1 −3
2
2
1
 0
3
6
0 −3 



4
4 
A =  2 −3 −2
.
 3 −6
0
6
5 
−2
9
2 −4 −5
24. Find a basis for the subspace of R4 spanned by the given vectors.
(a) (1, 1, −4, −3), (2, 0, 2, −2), (2, −1, 3, 2).
(b) (−1, 1, −2, 0), (3, 3, 6, 0), (9, 0, 0, 3).
(c) (1, 1, 0, 0), (0, 0, 1, 1), (−2, 0, 2, 2), (0, −3, 0, 3).
25. Find the rank and nullity of the matrix A.


1
4
5
6
9
 3 −2
1
4 −1 
.
A=
 −1
0 −1 −2 −1 
2
3
5
7
8
3
26. In each part use the information in the table to find the dimension of the row space of A, column space
of A, the null space of A.
Size of A
Rank(A)
(a)
3×3
3
(b)
3×3
2
(c)
3×3
1
(d)
5×9
2
(e)
9×5
2
(f)
4×4
0
(g)
6×2
2
27. Discuss how the rank of A varies with t.


1 1 t
(a) A =  1 t 1 .
t 1 1


t
3 −1
6 −2 .
(b) A =  3
−1 −3
t
28. Theorem. If ~x0 denotes any single solution of a consistent linear system A~x = ~b, and if vectors
{~v1 , ~v2 , · · · , ~vk } form a basis for the null space of A, that is, the solution space of the homogeneous
system A~x = ~0, then every solution of A~x = ~b can be expressed in the form
~x = ~x0 + c1~v1 + c2~v2 + · · · + ck~vk
(1)
and, conversely, for all choices of scalars c1 , c2 , · · · , ck , the vector ~x in this formula is a solution of
A~x = ~b.
Definition. In formula (1), the vector ~x0 is called a particular solution of A~x = ~b and the expression
~x = ~x0 + c1~v1 + c2~v2 + · · · + ck~vk ,
c1 , c2 , · · · , ck ∈ R
is called the general solution of A~x = ~b.
Using the above information, find the general solution of the system
x1
2x1
+ 3x2
+ 6x2
2x1
+
−
−
2x3
5x3
5x3
6x2
−
+
+
2x4
10x4
8x4
+ 2x5
+ 4x5
+
4x5
− 3x6
+ 15x6
+ 18x6
=
0
= −1
=
5
=
6
29. Suppose that x1 = −1, x2 = 2, x3 = 4, x4 = −3 is a solution of a nonhomogeneous linear system
A~x = ~b and that the solution set of the homogeneous system A~x = ~0 is given by the formulas
x1 = −3r + 4s, x2 = r − s, x3 = r, x4 = s,
(a) Find the general solution of A~x = ~0.
(b) Find the general solution of A~x = ~b.
4
r, s ∈ R.