Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Digital systems, linear algebra, introduction to formal proof / 5 / b

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- Solution b

Solution

Write the columns of

A

a_{1}, \dots, a_{n}

and the coordinates of the vector

x

x_{1}, \dots, x_{n}

. By the definition of matrix multiplication,

y = A x = x_{1} a_{1} + \dots + x_{n} a_{n} \in span {a_{1}, \dots, a_{n}} = Col (A) .

Indeed, its

i

th coordinate is

\sum_{j = 1}^{n} A_{ij} x_{j}

, exactly the

i

th coordinate of this linear combination. Since the column space contains every linear combination of the columns, the claim holds for every vector

x

.

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