Bases

Definition 1.4.1 A finite set S of elements in a linear space V is called a finite basis for V if S is independent and spans V. The space V is called finite-dimensional if it has a finite basis, or if V consists of 0 alone. Otherwise, V is called injinite-dimensional.

Theorem 1.4.3 (Uniqueness of Basis Representation) Every v ∈ V can be written uniquely as linear combination i.e.: ∀v ∈ V there exist n scalars, λ₁,...,λ_n ∈ 𝕂 such that

Proof. Suppose there is another linear combination for the same vector v as

by subtracting the two we have, 0 = ∑ⁿ_i=0 (λ_i − μ_i)v_i, which implies, owing to the independence of the vectors λ_i = μ_i ∀i.   □

The coefficients λ₁, λ₂, ..., λ_n are called scalar components of v with respect to the basis {v₁, v₂, ..., v_n}. For a specified basis we can write the vector v as

$$\mathbf{v}=({\lambda }_1,{\lambda }_2,\dots {\lambda }_n),\text{or}\mathbf{v}=\left(\begin{array}{c}{\lambda }_1\\ {\lambda }_2\\ ⋮\\ {\lambda }_n\end{array}\right)$$

we call the notation on the left row vector and that on the right column vector. We shall use mostly columns vectors, basically to multiply them to the left by matrices. Obviously changing the basis components change.

1.4.4 Examples.

• In 𝕂ⁿ the vectors

  e₁ = (1, 0, 0, ..., 0)
  

  e₂ = (0, 1, 0, ..., 0)
  

  ...
  

  e_n = (0, 0, 0, ..., 1)
  

  represent a basis called the standard (or canonical).

• The list 1, x, ..., xⁿ a basis of P_n(K) of polynomials of degree n or less. Consider for example the polynomial

  p = ax² + bx + c

  the three polynomials

  e₁ = x²   e₂ = x   e₃ = 1

  which are clearly linear independent, made a basis of V. Thus V has n = 3 as dimension.

1.4.5. Definition. A maximal linearly independent set in V is a linearly independent set 𝓑 in V such that if a vector v in V but not in 𝓑 is added to 𝓑, then the new set is linearly dependent.

1.4.6. Proposition. Let 𝓑 = {v₁, v₂, ..., v_n} a basis of a vector space V, then 𝓑 is a maximal set of linear independent vectors of V.

Proof. By definition the vectors {v₁, v₂, ..., v_n} are linear independent; we must show that for every v ∈ V, the vectors v₁, v₂, ..., v_n are linear dependent. Since 𝓑 spans V, there exist λ₁,...,λ_n ∈ 𝕂, such that

thus we have

which is a linear dependence relation between the vectors v₁, v₂, ...,v_n (the coefficient of v is 1, which is not null).   □

Thus, it is clear that a basis of a vector space is a maximal linearly independent set of vectors in that space. Because both the concepts of span and independence depend on the scalar field F, a basis too depends on F, as illustraed by the following example.

1.4.7. Example. Over the field of complex numbers, the vector space of complex numbers has dimension 1. A basis is {1}. Over the field of real numbers, the vector space of complex number has dimension 2. A basis is {1, i}.

«Spanning Sets and Linear Independence Index Definizione limite per funzioni operanti tra spazi metrici o normati »