Question

If we assume that the definition of the inner product of the normalized vectors `hatu_1` and `hatu_2` is `2u_1^2 + u_2^2` (these are vector components), how do you use the Gram-Schmidt process to generate orthonormal vectors from `vecv_1` and `vecv_2`?

`vecv_1 = (2,1)`
`vecv_2 = (5,-7)`

Answer 1

See below.

Explanation:

Following the Gram-Schmidt orthonormalization process, given `{vec u_1, vec u_2}` we begin by one of then, for instance `vec u_1` and define

`vec v_1 = (vec u_1)/norm(vec u_1)_q`

and then

`vec v_2 = (vec u_2 - << vec u_2,vec v_1 >>_q vec v_1)/norm( vec u_2 - << vec u_2,vec v_1 >>_q vec v_1)_q`

so if

`{(vec u_1=(2,1)),(vec u_2=(5,-7)):}`

we obtain

`{(vec v_1 =(2/3, 1/3) ),(vec v_2 =(sqrt 2/3, -(2 sqrt[2])/3) ):}`

NOTE: Here `<< cdot, cdot >>_q` applied to two vectors `vec u, vec v` gives `<< vec u, q cdot vec v >>` and

`norm(vec u)_q = sqrt(<< vec u, q cdot vec u >>)` with `q` a definite positive matrix.

In the present case `q = ((2,0),(0,1))`

Answer 2

I did your question from the beginning and got:

`hatu_1 = (2/3,1/3)`

`hatu_2 = (1/(3sqrt2), -4/(3sqrt2))`

---

The Gram-Schmidt process for two vectors first involves orthogonalizing them:

`vecu_1 = vecv_1`, `" "" "" "" "" "" "" "hatu_1 = vecu_1/||vecu_1||`

`vecu_2 = vecv_2 - "proj"_(vecu_1)vecv_2`, `" "" "hatu_2 = vecu_2/||vecu_2||`

And if you had more than two vectors `vecu_i`, you would match the subscript of `vecu_i` with `vecv_i`, then sequentially subtract the projection of the current `vecv_i` (highest index `i`) onto the previous `vecu_i`'s.

You have been given

• `vecv_1 = (2, 1)`
• `vecv_2 = (5, -7)`

Now, you already have `hatu_1`. Due to the definition of your inner product, the norm is not the usual definition (difference highlighted in red):

`|| vecu ||^2 = << vecu, vecu >> = color(red)(2)u_1u_1 + u_2u_2`

`=> color(blue)(hatu_1) = ((2", "1))/(sqrt(color(red)(2) xx (2)^2 + 1^2)) = ulcolor(blue)((2/3, 1/3)" ")`,

which indeed has a norm of `1` under YOUR inner product definition.

To proceed, we define the projection of `bb(vecv_2)` onto `bb(vecu_1)`:

`"proj"_(vecu_1)vecv_2 = (<< vecv_2, vecu_1 >>)/(<< vecu_1, vecu_1 >>) vecu_1`

How I remember it is that the vector that is projected (mapped), `vecv_2`, is the only different term in the projection definition.

We continue by finding the inner product of `vecv_2` with `vecu_1`, using YOUR definition, which is again, not quite the dot product of the vectors:

`<< vecv_2, vecu_1 >> = color(red)(2)v_(21)u_(11) + v_(22)u_(12)`

`= color(red)(2) xx 5 cdot 2 + -7 cdot 1`

`= 13`

And the inner product of `vecu_1` with itself is the norm squared:

`<< vecu_1, vecu_1 >> = ||vecu_1||^2 = 3^2 = 9`

As a result, the projection of `vecv_2` onto `vecu_1` is:

`"proj"_(vecu_1)vecv_2 = 13/9 cdot (2, 1)`

`= (26/9, 13/9)`

and so, the vector `vecu_2` would be found as:

`color(red)(vecu_2) = vecv_2 - "proj"_(vecu_1)vecv_2`

`= (5, -7) - (26/9, 13/9)`

`= (45/9, -63/9) - (26/9, 13/9)`

`= color(red)((19/9, -76/9)" ")`

Note that this is NOT normalized yet. The normalization of this is then:

`color(blue)(hatu_2) = (vecu_2)/(||vecu_2||)`

`= ((19/9", "-76/9))/sqrt(color(red)(2) xx (19/9)^2 + (-76/9)^2)`

`= (19/(19 sqrt2/3 cdot 9)", "-76/(19 sqrt2/3 cdot 9))`

`= ulcolor(blue)((1/(3sqrt2)", "-4/(3sqrt2))" ")`

And indeed this has a magnitude of `1` under this inner product definition.

Lastly, to check whether we are correct, we should see if the inner product of `hatu_1` and `hatu_2` is `0` (for orthogonality only). We already know these vectors are properly normalized in this particular inner product space.

`0 stackrel(?" ")(=) << hatu_1, hatu_2 >>`

`= color(red)(2) xx 2/3 cdot 1/(3sqrt2) + 1/3 cdot -4/(3sqrt2)`

`= 4/(9sqrt2) + (-4/(9sqrt2)) = 0` `color(blue)(sqrt"")`

So, this should be correct! (If you wish, you can check that `<< hatu_1, hatu_1 >> = << hatu_2, hatu_2 >> = 1`, which is again, `color(red)(2)u_(11)^2 + u_(12)^2`, and `color(red)(2)u_(21)^2 + u_(22)^2`, respectively.)