 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/digital-systems-linear-algebra-and-introduction-to-formal-proof/4/b/solution

A^{⊤} x = 0

, the vector

x

belongs to the null space for which the columns of

L

are a basis. It is therefore a linear combination of those columns, so

x = L y

for some

y \in R^{m}

. Conversely, each column is annihilated by

A^{⊤}

, hence

A^{⊤} L = 0

and

x = L y

implies

A^{⊤} x = 0

. Thus

A^{⊤} x = 0 ⟺ x = L y for some y \in R^{m} .

The coordinate vector

y

is unique because the columns of

L

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