Question

Convert the basis v1 = (-1,1,0)^T, V2 = (2,-1, 1)^T, V3 = (0, -1, 1)^T for Rinto an orthonormal basis, using the Gram Schmidt process and the standard inner product in R?

Answer 1

`{\bf e}_(1)=((-1,1,0))/(\lVert (-1,1,0) \rVert)=\left((-1)/(√(2)),(1)/(√(2)),0\right)`

`{\bf e}_(2)=\frac{{\bf v}_(2)-({\bf v}_(2)\cdot {\bf e}_(1)) {\bf e}_(1)}{\lVert {\bf v}_(2)-({\bf v}_(2)\cdot {\bf e}_(1)) {\bf e}_(1) \rVert}=\left((1)/(√(6)),(1)/(√(6)),\sqrt{(2)/(3)}\right)`

`{\bf e}_(3)=\frac{{\bf v}_(3)-({\bf v}_(3)\cdot{\bf e_(1)}){\bf e}_(1)-({\bf v}_(3)\cdot {\bf e}_(2)){\bf e}_(2)}{\lVert {\bf v}_(3)-({\bf v}_(3)\cdot{\bf e_(1)}){\bf e}_(1)-({\bf v}_(3)\cdot {\bf e}_(2)){\bf e}_(2) \rVert}=\left(-(1)/(√(3)),-(1)/(√(3)),(1)/(√(3))\right)`

Explanation:

We have the basis of
`\mathbb{R}^(3)`
`{\cal B}=\{(-1,1,0),(2,-1,1),(0,-1,1)\}`. From this basis we want to determine another orthonormal basis
`{\cal B}'=\{{\bf e}_(1),{\bf e}_(2),{\bf e}_(3)\}` of
`\mathbb{R}^(3)`.

The first step is to define
`{\bf e}_(1)` as:

`{\bf e}_(1)=((-1,1,0))/(\lVert (-1,1,0) \rVert)=\left((-1)/(√(2)),(1)/(√(2)),0\right)`

Now define
`{\bf e}_(2)` by:

`{\bf e}_(2)=\frac{{\bf v}_(2)-({\bf v}_(2)\cdot {\bf e}_(1)) {\bf e}_(1)}{\lVert {\bf v}_(2)-({\bf v}_(2)\cdot {\bf e}_(1)) {\bf e}_(1) \rVert}=\left((1)/(√(6)),(1)/(√(6)),\sqrt{(2)/(3)}\right)`

Now define
`{\bf e}_(3)` by :

`{\bf e}_(3)=\frac{{\bf v}_(3)-({\bf v}_(3)\cdot{\bf e_(1)}){\bf e}_(1)-({\bf v}_(3)\cdot {\bf e}_(2)){\bf e}_(2)}{\lVert {\bf v}_(3)-({\bf v}_(3)\cdot{\bf e_(1)}){\bf e}_(1)-({\bf v}_(3)\cdot {\bf e}_(2)){\bf e}_(2) \rVert}=\left(-(1)/(√(3)),-(1)/(√(3)),(1)/(√(3))\right)`