Vector Space

from University of Tokyo 1S2 Linear Algebra

Vector Space

• Definition: A set with defined addition and scalar multiplication.
  • Does this mean addition and scalar multiplication between elements of the set? (blu3mo)
• Axioms of a vector space:
  • Distributive law: $a\in R,v_1,v_2\in V⟹a(v_1+v_2)=a{v}_1+a{v}_2$
  • Various other axioms
  • (wikipedia)
    • Axioms are a list of the minimum necessary conditions.
      • Similar to the concept of linear independence (blu3mo)
      • Are there any other concepts like that?
  • There is no identity element in a vector space.
  • Assuming the existence of an external entity called a scalar, we define the relationship with scalars and addition between elements.
• What kind of thing is it?
  • $R^n$ is a typical example.
    • On the other hand, what else is there? (blu3mo)
      • Like (ℝ, Z)?
  • The set of functions is also a vector space.
    • If we define $a\cdot f(x)$ and $f(x)+g(x)$ in a sensible way, they satisfy the definition of a vector space.
    • I see. (blu3mo)
• Properties:
  • There are inverse elements and a zero element.
    • If we multiply an element by -1/0, we get the inverse/zero element.
  • Operations that can be described using only addition, scalar multiplication, negation, and zero hold in any vector space.
    • Well, that’s true, but I guess this is the joy of generalization. (blu3mo)
      • We can find symmetry by relating multiple specifics. (blu3mo)