Vectorial Spaces

Vectorial Spaces

1. We can add two vectors and we get another vector

2. We can multiply a vector by a number (scalar) and we get another vector.

3. Properties of the sum of vectors

   • Associative: (u + v) + w = ​​u + (v + w)
   • Commutative: v + u = u + v.

4. Properties of the product of a vector and a scalar.

   • Associative: β (α v) = (β α) v
   • ∙ Distributive:
     • Regarding the sum of scalars: (α + β) v = α v + β v
       • Regarding the sum of vectors: α (u + v) = α u + α v

5. Definition: vector space.

   • Any set that has sum and product operations by scalars, fulfilling all the previous properties, we will say that it is a vector space.

6. VECTOR SUBSPACES

   • Given a vector space V, a subset S of V is said to be a vector subspace if it contains the vector 0, and if when performing the operations of addition and product by scalar between vectors of S, the result remains in S.
     • SUBSPACE OPERATIONS
       • From two subspaces we can build another one by performing the operations of addition or intersection of subspaces.

7. Relationship between the implicit and parametric form.

   • No. of implicit equations + No. of parameters = n.

8. Inclusion of subspaces.

   • Given two subspaces A and B, it may happen that one is included in another
     • -Implicitly
     • -In parametric form:

9. Observation

   • The intersection S∩T is a subspace “smaller” than S and T (it is contained in S and also in T).

10. DEPENDENCE AND INDEPENDENCE

    • LINEAR Definition: Linear combination.
      • Given a set of vectors v1,. . ., vn any vector of the form v = α1 v1 +. . . + αn vn a linear combination of them is called
    • Theorem.
      • Para probar que un conjunto S es subespacio, basta probar que: La combinación lineal de elementos de S está en S.

11. Linear dependency.

    • A set of vectors is said to be linearly dependent (or linked) if: (a) At least one of them is a linear combination of the others. This can also be expressed like this:

12. GENERATOR SYSTEMS

    • Generated subspace and generator system.
      • Given a set of vectors v1,. . . , vr, the set of all possible linear combinations of them is called subspace generated (or generated) by v1,. . . , vr. It is also said that these vectors are a generator system of subspace (or space, where appropriate).

13. Reduction of generating systems.

    • If from a generating system we suppress the vectors that are a linear combination of the others, the remaining ones continue to generate the same subspace.