 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof/6/b/ii/solution

f : W_{1} \cap W_{2} \to U

f (u) = (u, - u)

. It is well-defined by the definition of

U

, and

f (a u + b v) = (a u + b v, - a u - b v) = a f (u) + b f (v) .

Thus

f

f (u) = f (v)

, comparing first components gives

u = v

, proving injectivity. Every element of

U

has the form

(u, - u) = f (u)

, proving surjectivity. Hence $f$ is a linear isomorphism, with inverse

(u, - u) \mapsto u

, and

dim U = dim (W_{1} \cap W_{2})

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