 Solution

ID: 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/solution

Write the columns of

A

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

and the coordinates of the vector

x

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

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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