(a) Let $B$ be an $n \times n$ positive-definite, symmetric matrix. Define the Cholesky factorization of $B$ and prove that it is unique.

(b) Let $A$ be an $m \times n$ matrix, $m \geqslant n$, such that $\operatorname{rank} A=n$. Prove the uniqueness of the "skinny QR factorization"

$$A=Q R,$$

where the matrix $Q$ is $m \times n$ with orthonormal columns, while $R$ is an $n \times n$ upper-triangular matrix with positive diagonal elements.

[Hint: Show that you may choose $R$ as a matrix that features in the Cholesky factorization of $B=A^{T} A$.]