Векторное пространство, span и базис
Векторное пространство, span и базис
Ответить самому
Сначала сформулируйте ответ как на собеседовании, затем откройте разбор и оцените себя.
Короткий ответ
A vector space supports vector addition and scalar multiplication with the usual axioms. A span is all linear combinations of a set of vectors; a basis is a linearly independent set whose span is the whole space.
Полный разбор
A vector space is a set of objects where you can add two vectors and multiply a vector by a scalar while satisfying closure, associativity, distributivity, identity and inverse properties.
The linear span of vectors v1...vk is the set of all linear combinations a1 v1 + ... + ak vk. It can be the whole space or a subspace.
A basis is a stronger object. It spans the target space and its vectors are linearly independent. Linear independence means no basis vector can be represented as a linear combination of the others. Because of that, every vector in the space has a unique coordinate representation in that basis.
Теория
Span gives coverage; basis gives coverage without redundancy.
Типичные ошибки
- Say any spanning set is a basis.
- Forget linear independence.
- Define basis only as coordinates without the spanning property.
Как отвечать на собеседовании
- Use R2: two non-collinear vectors form a basis, three vectors span but are dependent.