 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/digital-systems-linear-algebra-introduction-to-formal-proof-resit/6/a/solution

For a real

m \times n

A

, the row space is the linear span of its rows in

R^{n}

. The column space is the linear span of its columns in

R^{m}

, equivalently

{A x : x \in R^{n}}

ker A = {x \in R^{n} : A x = 0} .

The matrix rank is the dimension of the column space, which equals the dimension of the row space. In particular,

rank A = dim Row (A) = dim Col (A), dim ker A = n - rank A .

The last equality is the rank-nullity theorem.

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