Vector Space

Vector Space, $\chi$

Description:

• Obtained by equipping vectors in a certain dimension with their Linear Combination
  • ie, the possible space that the vectors can be

Properties:

• Closure under addition: 2 vectors adding cant escape the vector space
• Closure under scalar multiplication: 2 vectors multiply cant escape

Basis of a vector space:

• A set of least linearly independent vectors that Span the full space
  • ie, n vectors that will let it span the whole subspace
  • ex, basis vector of a line is 1 vector; basis vectors of a plane are 2 vectors
• If we have a basis $\{x^{(1)},...,x^{(m)}\}$ for a subspace, $S$, then we can uniquely write any element in the subspace as a Linear Combination of elements in that basis.
  • That is, any $x\in S$ can be written as $x=\sum _{i=1}^d{\alpha }_i{x}^{(i)}$, for an appropriate $\alpha$
• To prove:
  • 2 conditions

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