# Functionals and conjugate bases

For parameters $s,t\in\mathbb{R}$ vectors are given $\alpha_1=[1,1,0],\alpha_2=[1,1,s]$ from $\mathbb{R}^3$ and functional $f_1([x_1,x_2,x_3])=tx_1-x_2-x_3$ and $f_2([x_1,x_2,x_3])=-x_1+x_2+sx_3$ from $(\mathbb{R}^3)^*$

Now i need to investigate for which values of parameters s and t you can find a vector $\alpha_3 \in \mathbb{R}^3$ and functional $f_3\in(\mathbb{R}^3)^*$ such that system of equations $A=(\alpha_1, \alpha_2, \alpha_3)$ is a basis of $\mathbb{R}^3$ and $(f_1,f_2,f_3)$ is a basis of $(\mathbb{R}^3)^*$ conjugate to A

Detailed explanition needed, cause i do not get the question at all

Hint: We have $f_2(\alpha_1)=0, f_2(\alpha_2)=s^2$ and $f_1(\alpha_1)=t-1, f_1(\alpha_2)=t-1-s$. It implies $s=1,t=2.$