Compute $$\begin{vmatrix} 1 & x & x^2 \cr x & 1 & x^2 \cr x^2 & x & 1 \end{vmatrix}$$
Prove that in $\mathbb{K}[x]$ polynomials of degree $n$ do not form a subspace, but polynomials of degree $\leq n$ do.
Find a linear transformation $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that the volume of the 3-D parallelogram (parallelopiped) defined by $T(1,0,0), T(0,1,0),$ and $T(0,0,1)$ is $\frac{1}{2}$.