Linear Spaces

Vector spaces- A set L of elements x̅ , y̅ , z̅ , … are called linear space or vector space (on ℝ)
if addition and multiplication by scalars are defined so that the following laws are satisfied for all x̅ , y̅ , z̅ ∈ L and λ, μ ∈ ℝ.
    I.   (i) x̅ , y̅ ∈ L ⇒ x̅  +  y̅ ∈ L

         (ii) x̅  +  y̅ =  y̅ + x̅ 

         (iii) (x̅  +  y̅) +  z̅ = x̅  +  (y̅ +  z̅)

         (iv) ∃ 0̅ : x̅   + 0̅ = x̅  

         (v) ∃ -x̅  : x̅  + (-x̅ ) = 0̅

   II.  (i) λx̅  ∈ L

         (ii) λ(μx̅ ) = (λμ)x̅ 

         (iii) (λ + μ)x̅  = λx̅  + μx̅ 

         (iv) λ(x̅  +  y̅) = λx̅  + λ y̅

         (v) 1x̅  = x̅ 

         (vi) 0̅x̅  = 0̅

         (vii) λ0̅ = λ

Test for subsequence- A non-empty subset M of L is a linear space itself, if-

1. x̅ , y̅ ∈ M ⇒ x̅  +  y̅ ∈ M
2. x̅  ∈ M, λ ∈ ℝ ⇒ λx̅  ∈ M

Linear combinations,basis-

1. The vector  y̅  is a linear combination of vectors x̅_{1 , … ,} x̅_n, if,           y̅ = λ ₁ x̅_{1 +} λ ₂ x̅_{2 + … +} λ _n x̅_n, for some scalars λ _{1, … ,} λ _n.
2. The linear hull LH(x₁, …, x_n) is                                                                {y : y = λ ₁x_{1 +} λ ₂x_{2 + … +} λ _nx_n, λ _i∈ ℝ }
3. Vectors  x̅_{1 , … ,} x̅_n are  (a) Linearly independent, if                 λ ₁ x̅_{1 +} λ ₂ x̅_{2 + … +} λ _n x̅_n =  0̅                                                         ⇒ λ _i = 0, all i
        (b) Linearly dependent, if ∃ λ _{1, … ,} λ _n not all zero :
                        λ ₁ x̅_{1 +} λ ₂ x̅_{2 + … +} λ _n x̅_n =  0̅
                 (⇔ some x̅_i is a linear combination of the other)
4. e̅_1, e̅₂, ... , e̅_n is a basis of the linear space L and L is n-dimensional, if-

          (i) e̅_1, e̅₂, ... , e̅_n are linearly independent.

          (ii) Every x̅  ∈ L can be written uniquely,

             x =x₁e̅₁ +x₂e̅₂ + ... + x_ne̅_n

Scalar Product:

• Let L be a limear space, A scalar  product (x̅, y̅) is a function L × L → ℝ with the following prperties holding for all x̅, y̅, z̅ ∈ L and λ, μ ∈ ℝ-

       (a) (x̅, y̅) = (y̅, x̅)
       (b) (x̅, λy̅ + μz̅ ) = λ(x̅, y̅) + μ(x̅, z̅)
       (c) (x̅, x̅) ≥ 0, (x̅, x̅) = 0 ⇔ x̅ = 0

• Length of x̅  : |x̅ | = √(x̅, x̅),

                           |cx̅ | = (c)|x̅ | (c scalar)

• |(x̅, y̅)| ≤ |x̅ | |y̅ | (Cauchy- Schwarz inequality).
• |x̅ + y̅)| ≤ |x̅ | + |y̅ | (Triangle inequality).

Annihilating Polynomials:

Minimal polynomial- If T is a linear operator on a finite dimensional vector space V over the field F. The minimal polynimial for T is the (unique) monic generator of the ideal of polynomials over F which annihilate T.

The Transpose of a Linear Transformation:

• let V and W be vector space over the field F. For each linear transformation T from V into W, there is a unique linear transformation T^t from W* into V* such that 

                       (T^tg)(α) = g(Tα)

          for every g ∈ W* and α ∈ V.

• Let V and W be vector spaces over the field F, and let T be a linear transformation from V into W. The null space of T^t is the annihilator of the range of T. Then V and W are finite dimensional, then

          (i) rank (T^t) = rank (T)

         (ii) the range of T^t is annihilator of the null space of T.

• If V and W are finite dimensional vector space over the field F. B is an ordered basis for V with dual basis B*, B' is an ordered basis for W with dual basis B'*, T is a linear transformation from V into W. A and B are matrices of T and T' relative to B, B' and B'*, B* respectively. Then B_ij = A_ji.

Invariant Subspaces: Suppose V is a vector space and T a linear operator on V. If W is a subspace of V, then W is invariant subspace of V under T if for each vector α ∈ W, the vector Tα ∈ W
i.e.,  if T(W) is contained in W.

T Conductor of α into W- If W is a invariant subspaces for T and α is a vector in V. The T conductor of α into W is the set S_T (α : W), which consists of all polynomials g (over the scalar field) such that g(T) α is in W.

Cyclic-subspaces generated by α- If α is any vector in V, the T-cyclic subspace generated by α is the subspace Z(α : T) of all vectors of the form g(T), α,g ∈ F[x].
     If Z(α : T) = V then α is called a cyclic vector for T.