# Gram Schmidt Process Using Orthonormal Vectors

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

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

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

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

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

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

$$\frac {\langle v_2, u_1 \rangle } {\|u_1\|}=\langle v_2, \frac {u_1} {\|u_1\|} \rangle$$ 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 $$e_1$$.