I am studying Linear Algebra and have been introduced to Gram-Schimdt algorithm and I am struggling on how to approach this question.
For the beginning inner product I'm unsure how you get the answer:
$u_{1}w_{1}+2u_{2}w_{2}+3u_{3}w_{3}$
as from my linear algebra notes it only gives me the information to obtain:
$u_{1}w_{1}+u_{2}w_{2}+u_{3}w_{3}$ and I dont know how to get the missing coefficients, but assume this is due to these vectors being in $R^{3}$.
Also next for the Gram-Schmidt algorithm, I believe that I have to use the inner product.
I have found from a similar post that I should get:
$\|(1,1,1)\|=\sqrt6$ and that the first vector we should get when applying Gram-Schmidt is $e_1=\frac1{\sqrt6}(1,1,1)$
And then we let:
$a_2=(1,0,1)-\langle(1,0,1),e_1\rangle e_1=\left(\frac13,-\frac23,\frac13\right)$
Its norm is $\sqrt{\frac43}$ and so we take $e_2=\frac1{2\sqrt3}(1,-2,1)$
And finally I'm supposed to take $a_3=(0,1,2)-\langle(0,1,2),e_1\rangle e_1+\langle(0,1,2),e_2\rangle e_2=\left(-\frac32,0,\frac12\right)$
Its norm is $\sqrt3$ and so im supposed to take $e_3=\frac1{\sqrt3}\left(-\frac32,0,\frac12\right)$
But I cant understand what is going on here and was wondering if anyone could explain what is happening here and how to obtain the orthonomal basis from this.
Thanks in advance.
The algorithm of Gram-Schmidt is valid in any inner product space. If ${v_1,...,v_n}$ are the vectors that you want to orthogonalize (they need to be linearly independent otherwise the algorithm fails) then: $$w_1=v_1$$ $$w_2=v_2-\frac{\langle v_2,w_1 \rangle}{\langle w_1,w_1 \rangle}w_1$$ $$w_3=v_3-\frac{\langle v_3,w_1 \rangle}{\langle w_1,w_1 \rangle}w_1 -\frac{\langle v_3,w_2 \rangle}{\langle w_2,w_2 \rangle}w_2$$ $$\\.\\.\\.\\$$ $$w_j=v_j-\sum_{i=1}^{j-1} \frac{\langle v_j,w_i \rangle} {\langle w_i,w_i \rangle}w_i$$ $$\\.\\.\\.\\$$ $$w_n=v_n-\sum_{i=1}^{n-1} \frac{\langle v_n,w_i \rangle} {\langle w_i,w_i \rangle}w_i$$
Simply apply the algorithm. The intuition behind the algorithm is that to make the vector $v_j$ orthogonal to $w_1,...,w_{j-1}$ you have to make sure that its projections on $w_1,...,w_{j-1}$ are $0$(a vector is orthogonal to another if the projection of the former on the latter is $0$). So you subtract from $v_j$ its projections on $w_1,...,w_{j-1}$.