Suppose we have $n$ linearly independent vectors $\mathbf{x}_1$, $\cdots$, $\mathbf{x}_n$ in $\mathbb{R}^n$. Let $\mathbf{X}$ be the $n \times n$ matrix with column $k$ given by $\mathbf{x}_k$, $k = 1, \cdots, n$. I know that the volume of the parallelepiped defined by the set $$ E = \{\mathbf{x} \in \mathbb{R}^n: \mathbf{x} = \mathbf{X}\theta \text{ for some $\theta \in [0,1]^n$}\} $$ is given by $|\det \mathbf{X}|$, the absolute value of the determinant of $\mathbf{X}$.
How can I show that the Lebesgue measure of $E$ is also equal to $|\det \mathbf{X}|$?