 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2005/calculus-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/1/c/solution

A basis provides unique coordinates. Define

α (a_{1} e_{1} + a_{2} e_{2} + a_{3} e_{3}) = a_{1} v_{1} + a_{2} v_{2} + a_{3} v_{3} .

The uniqueness of the coordinates makes this a well-defined function. Adding coordinates and multiplying them by a real scalar shows directly that

α (a u + b w) = a α (u) + b α (w)

, so it is a linear map. Substituting

e_{i}

gives

α (e_{i}) = v_{i}

Conversely, every linear map with those three values must send

\sum_{i} a_{i} e_{i}

\sum_{i} a_{i} v_{i}

, by preservation of linear combinations. This determines its value on every element of

V

v_{1}, v_{2}, v_{3}

is needed.

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