 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof/4/a/ii/solution

Suppose a vector

v

lies outside the span of

S

. Then

S \cup {v}

αv + \sum_{i} β_{i} s_{i} = 0

, a nonzero

α

would express

v

in that span, and with

α = 0

the independence of

S

forces every

β_{i} = 0

This would give

n + 1

n

-vector basis, contrary to the supplied claim. Therefore

S

V

. Independence plus spanning makes $S$ a basis.

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