LU Decomposition, linear equations

The previous equation can be calculated in two steps $$\mathbf{L} \mathbf{y} = \mathbf{w};\qquad \mathbf{Ux}=\mathbf{y}.$$

To show that this is correct we use to the LU decomposition to rewrite our system of linear equations as $$\mathbf{LUx}=\mathbf{w},$$ and since the determinat of $\mathbf{L}$ is equal to 1 (by construction since the diagonals of $\mathbf{L}$ equal 1) we can use the inverse of $\mathbf{L}$ to obtain $$\mathbf{Ux}=\mathbf{L^{-1}w}=\mathbf{y},$$ which yields the intermediate step $$\mathbf{L^{-1}w}=\mathbf{y}$$ and as soon as we have $\mathbf{y}$ we can obtain $\mathbf{x}$ through $\mathbf{Ux}=\mathbf{y}$.

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