7.1 Orthogonal and orthonormal bases

what is an orthoganal set

• a set of vectors $[{\vec{v}}_1,{\vec{v}}_2,\dots ,{\vec{v}}_k]$is called an orthogonal set if ${\vec{v}}_i\cdot {\vec{v}}_j=0$ for every $i=/=j$

  • if in addition each $\mid \mid {\vec{v}}_i\mid \mid =1$ (each vector has length 1) the set is called an orthonormal set

• Basically each dot product pair needs to equal zero

what is an orthogonal set

• a set of vectors $[{\vec{v}}_1,{\vec{v}}_2,\dots ,{\vec{v}}_k]$is called an orthogonal set if ${\vec{v}}_i\cdot {\vec{v}}_j=0$ for every $i=/=j$

  • if in addition each $\mid \mid {\vec{v}}_i\mid \mid =1$ (each vector has length 1) the set is called an orthonormal setup

• Basicly each dot product pair needs to equal zero

• AN orthogonal set is always a linearly independent set

  • so if an orthogonal set B spans a subspace W, we say that B is an orthogonal basis for W.
  • similarly, an orthonormal set that spans W is called an orthonoramal basis for W.

Standard basis is an orthonormal basis for ${\mathbb{R}}^n$

• the standard basis $[{\vec{e}}_1,{\vec{e}}_2,\dots ,{\vec{e}}_n]$ is an orthonormal basis for ${\mathbb{R}}^n$

  • we can convert non orthonormal basis to an orthonormal basis

    • by dividing the each vector component by the length of it