Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 1 / a / i

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- Solution i

Solution

Write

A = (a_{ij})

as an

m \times r

matrix and

B

as an

r \times n

matrix, with rows

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

. The rule for matrix multiplication gives

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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