Applying Gram-Schmidt Process to transform set of vectors

Question

I have a question that goes as follows:

And below is my written solution to the question:

https://i.stack.imgur.com/CF05T.jpg

I just wanted to know if I've approached this question correctly and whether or not my solution is actually correct.

Answer 1

We have $<1,1>=2,<1,x>=0,<x,x>=\dfrac23$.

From this,

$$<1+x,1+x>=2+2\cdot0+\frac23=\frac83$$

$$<1+3x,1+3x>=2+6\cdot0+9\frac23=8$$

$$<1+x,1+3x>=2+4\cdot0+3\frac23=4$$

and your solution is not right.

Answer 2

Normalize the first element. Then remove the projection of the second vector along the first vector, and normalize that result: $$ e_1 = \frac{1}{\|u_1\|}u_1 \\ e_2 = \frac{u_2-\langle u_2,e_1\rangle e_1}{\|u_2-\langle u_2,e_1\rangle e_1\|} $$ This is the standard Gram-Schmidt process.