Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 6 / a / i / Solution

For independent columns $a_1,\dots ,a_k$ in a real inner product space, the Gram-Schmidt process recursively formsEach $u_j$ subtracts the orthogonal projections onto earlier normalized vectors. Their independence ensures $u_j\ne 0$. The resulting $q_i$ are orthonormal and preserve the span of every initial column set. Putting $Q=(q_1\cdots q_k)$ and $r_{ij}=⟨q_i,a_j⟩$ for $i\le j$, with zeros below the diagonal, gives a QR decomposition $A=QR$ with $Q^{T}Q=I$. Dependent input columns require omitting a zero residual or treating rank deficiency separately.