 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/digital-systems-linear-algebra-introduction-to-formal-proof/5/b/solution

x \in Null (A)

, the

i

th entry of

A x = 0

says

r_{i} \cdot x = 0

. Any

y \in Row (A)

y = \sum_{i} α_{i} r_{i}

y \cdot x = \sum_{i} α_{i} (r_{i} \cdot x) = 0.

Hence

Row (A) ⊥ Null (A) .

This proves the orthogonality for every pair of vectors in the two vector subspaces. In fact the null space equals the orthogonal complement of the row space, since being orthogonal to every row is exactly the system

A x = 0

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