Let's have vectors $v_1,v_2,v_3,v_4\in \mathbb R^4$. Prove that these vectors form basis (1), determine if they are orthogonal and/or orthonormal (2). Then make a transition matrix $T_{\epsilon\alpha}$ from basis $\alpha$ to standard basis and use it to get coordinates of vector $w=(2,3,5,1)_\alpha$ in standard basis (3). $$v_1=(1,1-1,1),v_2=(1,-1-1,-1),v_3=(0,1,0,-1),v_4=(1,0,1,0) $$ (1) Prove that these vectors form a basis.
I did following: $$\left( \begin{array}{cccc} 1 & 1 & -1 &1 \\ 1 & -1 & -1 &-1 \\ 0 & 1 & 0 &-1 \\ 1 & 0 & 1 & 0 \end{array} \right)=A$$ $$\left| \begin{array}{cccc} 1 & 1 & -1 &1 \\ 1 & -1 & -1 &-1 \\ 0 & 1 & 0 &-1 \\ 1 & 0 & 1 & 0 \end{array} \right|=-8 \ $$ $|A| \neq 0 \implies$ Vectors $v_1,v_2,v_3,v_4$ are linearly independent and form basis.
(2) Determine if they are orthogonal and/or orthonormal.
I did following: $$v_1\cdot v_2=(1-1+1-1)=0$$ $$v_1\cdot v_3=(0+1+0-1)=0$$ $$v_1\cdot v_4=(1+0-1+0)=0$$ $$v_2\cdot v_3=(0-1+0+1)=0$$ $$v_3\cdot v_4=(0+0+0+0)=0$$ $\implies$ Basis is orthogonal. $$|v_1|=\sqrt {1+1+1+1}=\sqrt4=2$$ $$|v_2|=\sqrt {1+1+1+1}=\sqrt4=2$$ $$|v_3|=\sqrt {0+1+0+1}=\sqrt2$$ $$|v_4|=\sqrt {1+0+1+0}=\sqrt2$$ $\implies$ Basis is not orthonormal.
(3) Make a transition matrix $T_{\epsilon\alpha}$ from basis $\alpha$ to standard basis and use it to get coordinates of vector $w=(2,3,5,1)_\alpha$ in standard basis.
$$T_{\epsilon\alpha}=A^{-1}=\left( \begin{array}{cccc} \frac14 & \frac14 & 0 & \frac12 \\ \frac14 & -\frac14 & \frac12 &0 \\ -\frac14 & -\frac14 & 0 & \frac12 \\ \frac14 & -\frac14 & -\frac12 & 0 \end{array} \right)$$
$$u=A^{-1} w$$ Where $u$ is in standard basis?
Is (1) and (2) correct and can you help me out with (3) ?