Linear Algebra and Analysis

3.3 Exercises

1. Show that the following are subspaces of the set of all functions defined on
   [a,b]
   .
   1. polynomials of degree ≤ n
   2. polynomials
   3. continuous functions
   4. differentiable functions
2. Show that every subspace of a finite dimensional vector space V is the span of some vectors. It was done above but go over it in your own words.
3. In ℝ² define a funny addition by
   (x,y)
   +
   (ˆx,ˆy)
   ≡
   (3x+ 3ˆx,y +yˆ)
   and let scalar multiplication be the usual thing. Would this be a vector space with these operations?
4. Determine which of the following are subspaces of ℝ^m for some m. a,b are just given numbers in what follows.
   1. {(x,y) ∈ ℝ2 : ax + by = 0}
   2. { 2 } (x,y) ∈ ℝ : ax + by ≥ y
   3. { } (x,y) ∈ ℝ2 : ax + by = 1
   4. {(x,y) ∈ ℝ2 : xy = 0}
   5. { 2 } (x,y) ∈ ℝ : y ≥ 0
   6. { } (x,y) ∈ ℝ2 : x > 0 or y > 0
   7. For those who recall the cross product,

      { 3 } x ∈ ℝ : a × x = 0

   8. For those who recall the dot product,

      {x ∈ ℝm : x ⋅a = 0}

   9. {x ∈ ℝn : x ⋅a ≥ 0}
   10. m m {x ∈ ℝ : x ⋅s = 0 for all s ∈ S,S ⁄= ∅,S ⊆ ℝ }
       . This is known as S^⊥.
5. Show that
   { (x,y,z) ∈ ℝ3 : x+ y − z = 0}
   is a subspace and find a basis for it.
6. In the subspace of polynomials on
   [0,1]
   , show that the vectors
   { } 1,x,x2,x3
   are linearly independent. Show these vectors are a basis for the vector space of polynomials of degree no more than 3.
7. Determine whether the real valued functions defined on ℝ

   {x2 + 1,x3 + 2x2 + x,x3 + 2x2 − 1,x3 + x2 + x}

   are linearly independent. Is this a basis for the subspace of polynomials of degree no more than 3? Explain why or why not.

8. Determine whether the real valued functions defined on ℝ

   { } x2 + 1,x3 + 2x2 + x,x3 + 2x2 + x,x3 + x2 + x

   are linearly independent. Is this a basis for the subspace of polynomials of degree no more than 3? Explain why or why not.

9. Show that the following are each a basis for ℝ³.
   1. ( ) 3 |( 2 |) − 1
      ,
      ( ) 2 |( 2 |) − 1
      ,
      ( ) − 1 |( − 1 |) 1
   2. ( ) − 2 |( 0 |) 2
      ,
      ( ) 3 |( 1 |) − 2
      ,
      ( ) 4 |( 1 |) − 2
   3. ( ) − 3 |( 0 |) 1
      ,
      ( ) 5 |( 1 |) − 1
      ,
      ( ) 6 |( 1 |) − 1
   4. ( ) 1 |( 2 |) − 1
      ,
      ( ) 2 |( 2 |) − 1
      ,
      ( ) − 1 |( − 1 |) 1
10. Show that each of the following is not a basis for ℝ³. Explain why they fail to be a basis.
    1. ( ) | 1 | ( 1 ) 1
       ,
       ( ) | 0 | ( 1 ) 1
       ,
       ( ) | 3 | ( 5 ) 5
    2. ( ) | 1 | ( − 1 ) 1
       ,
       ( ) | 0 | ( 1 ) 1
       ,
       ( ) | 3 | ( − 1 ) 5
    3. ( ) 3 |( 2 |) 5
       ,
       ( ) 0 |( 1 |) 1
       ,
       ( ) 1 |( 0 |) 1
    4. ( ) 1 |( 0 |) 1
       ,
       ( ) 1 |( 1 |) 0
    5. ( ) 1 |( 2 |) − 1
       ,
       ( ) 2 |( 2 |) − 1
       ,
       ( ) − 1 |( − 1 |) 1
       ,
       ( ) 1 |( 0 |) 0
11. Suppose B is a subset of the set of complex valued functions, none equal to 0 and defined on Ω and it has the property that if f,g are different, then fg = 0. Show that B must be linearly independent.
12. Suppose you have continuous real valued functions defined on
    [0,1]
    ,
    {f1,f2,⋅⋅⋅,fn}
    and these satisfy

    { ∫ 1 1 if i = j 0 fi(x)fj(x)dx = δij ≡ 0 if i ⁄= j

    Show that these functions must be linearly independent.

13. Show that the real valued functions cos
    (2x)
    ,1,cos²
    (x)
    are linearly dependent.
14. Show that the real valued functions e^x sin
    (2x)
    ,e^x cos
    (2x)
    are linearly independent.
15. Let the field of scalars be ℚ and let the vector space be all vectors (real numbers) of the form a + b
    √2
    for a,b ∈ ℚ. Show that this really is a vector space and find a basis for it.
16. Consider the two vectors
    ( ) 2 1
    ,
    ( ) 1 2
    in ℝ². Show that these are linearly independent. Now consider
    ( ) 2 1
    ,
    ( ) 1 2
    in ℤ₃² where the numbers are interpreted as residue classes. Are these vectors linearly independent? If not, give a nontrivial linear combination which is 0.
17. Is ℂ a vector space with field of scalars ℝ? If so, what is the dimension of this vector space? Give a basis.
18. Is ℂ a vector space with field of scalars ℂ? If so, what is the dimension? Give a basis.
19. The space of real valued continuous functions on
    [0,1]
    usually denoted as C
    ([0,1])
    is a vector space with field of scalars ℝ. Explain why it is not a finite dimensional vector space.
20. Suppose two vector spaces V,W have the same field of scalars F. Show that V ∩ W is a subspace of both V and W.
21. If V,W are two sub spaces of a vector space U, define V + W ≡
    {v+ w : v ∈ V, w ∈ W }
    . Show that this is a subspace of U.
22. If V,W are two sub spaces of a vector space U, consider V ∪W, the vectors which are in either V or W. Will this be a subspace of U? If so, prove it is the case and if not, give an example which shows that it is not necessarily true.
23. Let V,W be vector spaces. A function T : V → W is called a linear transformation if whenever α,β are scalars and u,v are vectors in V , it follows that

    T (αu + βv) = αT u+ βT v.

    Then ker

    (T)
    ≡
    {u ∈ V : T u = 0}
    ,Im
    (T )
    ≡
    {T u : u ∈ V}
    . Show the first of these is a subspace of V and the second is a subspace of W.

24. ↑In the situation of the above problem, where T is a linear transformation, suppose S is a linearly independent subset of W. Define T⁻¹
    (S )
    ≡
    {u ∈ V : Tu ∈ S}
    . Show that T⁻¹
    (S)
    is linearly independent.
25. ↑In the situation of the above problems, rank
    (T )
    is defined as the dimension of Im
    (T )
    . Also the nullity of T, denoted as null
    (T)
    is defined as the dimension of ker
    (T)
    . In this problem, you will show that if the dimension of V is n, then rank
    (T )
    + null
    (T)
    = n.
    1. Let a basis for ker
       (T)
       be
       {z1,⋅⋅⋅,zr}
       . Let a basis for Im
       (T)
       be
       {Tv1,⋅⋅⋅,Tvs}
       . You need to show that r+s = n. Begin with u ∈ V and consider Tu. It is a linear combination of
       {Tv1,⋅⋅⋅,Tvs}
       say ∑ _i=1^sa_iTv_i. Why?
    2. Next explain why T
       ∑s (u− i=1aivi)
       = 0. Then explain why there are scalars b_j such that u −∑ _i=1^sa_iv_i = ∑ _j=1^rb_jz_j.
    3. Observe that V = span
       (z1,⋅⋅⋅,zr,v1,⋅⋅⋅,vs)
       . Why?
    4. Finally show that
       {z1,⋅⋅⋅,zr,v1,⋅⋅⋅,vs}
       is linearly independent. Thus n = r + s.