I have determined two vectors to be both orthogonal and linearly independent, but how do I change $u_2$ so that $u_1,u_2$ at orthonormal. So, $u_1=\begin{bmatrix} 0.6 \\0.8 \end{bmatrix}$,$u_2=\begin{bmatrix} 0.4 \\-0.3 \end{bmatrix}$.
My workings so far:
$(0.6)(0.4)+(0.8)(-0.3)= 0 $ so $||q_2|| \ne 1$ thus $u_!,u_2$ are orthogonal and linearly indepdent, but what changes to $u_2$ make these two vectors orthormal?
In general, we can normalize vectors by dividing their entries by the norm of the vector. In the general case, we normalize a vector $\textbf{a} =[a_1,a_2, ..., a_n]$ by
$$ \textbf{a} = \left[ \frac{a_1}{||a||},\frac{a_2}{||a||}, ..., \frac{a_n}{||a||} \right] $$
Where
$$ ||a|| = \sqrt{a_1^2+a_2^2 + ... + a_n^2} $$
So just normalize the vectors you have there and you will have orthonormal vectors.
i.e. $u_1$ becomes
$$ u_N = \left[\frac{0.6}{\sqrt{0.6^2+0.8^2}} , \frac{0.8}{\sqrt{0.6^2+0.8^2}}\right]^T $$
Where $u_N$ is the normalized unit vector of $u_1$.