0
$\begingroup$

Given $v_1,v_2,...,v_n$ vector the process is:

$u_1=v_1\Rightarrow e_1=\frac{u_1}{\|u_1\|}$

$u_2=v_2-\frac{<v_2,u_1>}{<u_1,u_1>}u_1\Rightarrow e_2=\frac{u_2}{\|u_2\|}$

And so on, I am trying to derive how can one use only the orthonormal vectors for example for $v_2$:

$$u_2=v_2-\frac{<v_2,u_1>}{<u_1,u_1>}u_1=v_2-\frac{<v_2,u_1>}{\|u_1\|^2}u_1=v_2-\frac{<v_2,u_1>}{\|u_1\|}\frac{u_1}{\|u_1\|}=v_2-\frac{<v_2,u_1>}{\|u_1\|}e_1$$

How can we go from $$\frac{<v_2,u_1>}{\|u_1\|}\Rightarrow <v_2,e_1>$$ Which properties of the inner product can we use?

$\endgroup$
1
$\begingroup$

$\frac {\langle v_2, u_1 \rangle } {\|u_1\|}=\langle v_2, \frac {u_1} {\|u_1\|} \rangle$ because $ \langle a, cb \rangle =c\langle a, b \rangle $ for $c$ real. [Here $c=\frac 1{\|u_1\|}$]. Now just use the defintion of $e_1$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.