Questions about determinants: their computation or their theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.
The determinant is a value that can be computed from the entries of a square matrix. This value is different from $0$ if and only if the matrix has an inverse and the determinant of the identity matrix is equal to $1$. For instance, for a $2\times 2$ matrix whose entries of the first line are $a$ and $b$ and whose entries of the second line are $c$ and $d$, the determinant is $ad-bc$.
If $f\colon\mathbb{R}^n\longrightarrow\mathbb{R}^n$ is a linear map and if $b$ is a basis of $\mathbb{R}^n$, then the determinant of the matrix of $f$ with respect to $b$ does not depend upon the choice of $b$; this number is called the determinant of $f$. The linear map $f$ has an inverse if and only if its determinant is not $0$.
Determinants are useful in the analysis of systems of linear equations and in the study of endomorphisms of finite-dimensional vector spaces.
