Inner product space computation 
If $x = (x_1,x_2)$ and $y = (y_1,y_2)$ show that $\langle x,y\rangle = \begin{bmatrix}x_1 & x_2\end{bmatrix}\begin{bmatrix}2 & -1 \\ 1 & 1\end{bmatrix}\begin{bmatrix}y_1 \\ y_2\end{bmatrix}$ defines an inner product on $\mathbb{R}^2$.

Is there any hints on this one? All I'm thinking is to compute a determinant, but what good is that? 

Use the definition of an inner product and check whether your function satisfies all the properties. Note that in general, as kahen pointed out in the comment, $\langle \mathbf{x}, \mathbf{y}\rangle = \mathbf{y}^*A\mathbf{x}$ defines an inner product on $\mathbb{C}^n$ iff $A$ is a positive-definite Hermitian $n \times n$ matrix.
