Abstract Algebra: Theory and Applications

What is a nonsingular matrix? Give an example of a $2\times 2$ nonsingular matrix. How do you know your example is nonsingular?

What is an isometry in ${\mathbb{R}}^n\text{?}$ Can you give an example of an isometry in ${\mathbb{R}}^2\text{?}$

What is an orthonormal set of vectors?

What is the difference between the orthogonal group and the special orthogonal group?

What is a lattice?