I have determined the following two vectors to be linearly independent. But how do I change $u_2$ so $u_1,u_2$ are orthonormal vectors?
My workings:
$u_1 = \begin{bmatrix} 1 \\0 \end{bmatrix}, $ $u_2 = \begin{bmatrix} 1 \\-1 \end{bmatrix}$ So, $(1)(-1)+(0)(1)=-1$thus these two vectors are linearly independent, but what can I change $u_2$ to make then orthonormal?
Just apply Gramm-Schmidt process to the set $\{u_1, u_2\}$.
We set $$\hat{v}_1 = u_1 = (1,0)$$ then $$v_1=\hat{v}_1/||\hat{v}_1|| = (1,0)$$ Now set $$\hat{v}_2=u_2-<u_2,v_1>v_1 = (1,-1) - 1\cdot(1,0) = (0,-1)$$ then $$v_2=\hat{v}_2/||\hat{v}_2|| = (0,-1)$$ So the vectors you're looking for are $$\{v_1,v_2\}= \{(1,0),(0,-1)\}$$