 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-i/1/a/i/solution

Write

A = (a_{ij})

as an

m \times r

B

as an

r \times n

matrix, with rows

b_{1}, \dots, b_{r}

row_{i} (A B) = \sum_{j = 1}^{r} a_{ij} b_{j} .

Every row of

A B

is therefore a linear combination of rows of

B

. Taking linear spans gives

rowspace (A B) \subseteq rowspace (B) .

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