Basis

Description:

• A set of least linear 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
  • There is no linear combination that can escape the subspace?
• If the $n$-vectors $a_1,...,a_k$ are linearly independent, then $k\le n$.
  • A linearly independent collection of $n$-vectors can have at most $n$ elements.
  • ie. Any collection of $n+1$ or more $n$-vectors is linearly dependent.

Orthonormal Basis:

• Orthogonality
• Same idea as Orthonormal Vector but the length of all are 1 and they are also able to reach all of Subspace
• A basis $(u_i{)}_{i=1}^n$ is said to be orthogonal if $u_i^T{u}_j=0$