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

Table of contents
- Solution 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}

to

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

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

V

, proving uniqueness. No linear independence of

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

is needed.

 Ancestors (11)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version