 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/a/solution

For an

m \times n

A

over the real numbers, write its rows as

r_{1}, \dots, r_{m}

and its columns as

c_{1}, \dots, c_{n}

. Then

Row (A) Col (A) Null (A) rank A = span {r_{1}, \dots, r_{m}} \subseteq R^{n}, = span {c_{1}, \dots, c_{n}} \subseteq R^{m}, = {x \in R^{n} : A x = 0}, = dim Row (A) = dim Col (A) .

Thus the row space and column space consist of all linear combinations of the corresponding rows and columns; the null space is the kernel of a linear map represented by

A

. The matrix rank counts independent row or column directions, with row rank equal to column rank.

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