1. ## Gram-Schmidt process question

Can anyone help me with this? I have to: apply the Gram-Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span(S), normalize it to obtain an orthonormal basis for span(S), and compute the Fourier coefficients of the given vector relative to the orthonormal basis. A lot of words, I know!

For some reason, the authors of my book thought it would be clever to let us (the students) figure all this out based on just the given theorems, and has not included any good examples. Anyway, here's the info for the problem:

V = R3
S = {(1,1,1), (0,1,1), (0,0,1)}
x = (1,1,2)

Any help would be MUCH appreciated!

2. Don't know what is doing that $x$ there, is it another vector?

3. It's for calculating the Fourier coefficients.

4. Originally Posted by paupsers
Can anyone help me with this? I have to: apply the Gram-Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span(S), normalize it to obtain an orthonormal basis for span(S), and compute the Fourier coefficients of the given vector relative to the orthonormal basis. A lot of words, I know!

For some reason, the authors of my book thought it would be clever to let us (the students) figure all this out based on just the given theorems, and has not included any good examples. Anyway, here's the info for the problem:

V = R3
S = {(1,1,1), (0,1,1), (0,0,1)}
x = (1,1,2)

Any help would be MUCH appreciated!
$S=\{v_{1}, v_{2}, v_{3}\}$

To find the orthogonal vectors, do the following...

$w_{1}=v_{1}$
$w_{2}=v_{2}-proj_{w_{1}}v_{2}$ as in $w_{2}=v_{2}-\frac{}{}w_{1}$
$w_{3}=v_{3}-proj_{w_{1}}v_{3}-proj_{w_{2}}v_{3}$

Now, normalize your $w_{i}$'s...

$u_{i}=\frac{w_{i}}{||w_{i}||}$

Then, the orthonormal basis you are looking for is the set of your $u_{i}$'s.

Keep in mind, $span(\{v_{1},v_{2},v_{3}\})=span(\{u_{1},u_{2},u_{ 3}\})$ .

That's the GOP.