Volumes

Unit $n$-cube

The unit n-cube is the $n$-dimensional cube with sides given by the standard basis vectors and lower-left corner located at the origin:

$C_n=\{\vec{x}\in\R^n:\vec{x}=\sum_{i=1}^n a_i\vec{e}_i \text{ for some } a_1, \dots , a_n \in [0,1]\}=[0,1]^n$

Determinant

The determinant of linear transformation $T:\R^n\to\R^n$ is $\text{det}(T)$ or $|T|$

= the oriented volume of the image of the unit $n$-cube.

The determinant of $n\times n$ matrix $M$ is $\text{det}(M)=\text{det}(T_M)$

Orientation-Preserving vs Reversing

• $T:\R^n\to\R^n$ is orientation-preserving if the ordered basis $\{T(\vec{e}_1),\dots,T(\vec{e}_n)\}$ is positively oriented.
• $T:\R^n\to\R^n$ is orientation-reversing if the ordered basis $\{T(\vec{e}_1),\dots,T(\vec{e}_n)\}$ is negatively oriented.
• If $\{T(\vec{e}_1),\dots,T(\vec{e}_n)\}$ is not a basis for $\R^n$, then $T$ is neither.

Oriented Volume: Positive if $T$ is orientation-preserving

Note: Determinants are only for square matrices or LT with domain = codomain!

Determinant Theorems

Let $T:\R^n\to\R^n,S:\R^n\to\R^n$ be linear transformations:

• $\det(S \circ T)=\det(S)\det(T)=\det(T \circ S)$

Let $A,B$ be $n \times n$ matrices:

• $\det(AB)=\det(A)\det(B)=\det(BA)$

For square matrix $M$, $\det(M)$ is the oriented volume of the parallelepiped ($n$-dimensional analog of a parallelogram) given by the column or row vectors.