7. Subspace of a Vector Space

Question 1

Subpace of Vector Space

Answer 1

A subset W of a vector space is called a subspace of V is W itself is a vector space with the same scalars, addition and scalar multiplication as V

Question 2

Spanning Set of Vector Space

Answer 2

Let V be a subset of a vector space, the set is called a spanning set of V if every vector in V can be written as a linear combination of vectors in S.

Question 3

Linear Independence

Answer 3

The only solution to the linear combination is 0.

Question 4

Basis of Vector Space

Answer 4

A subset B of an F-vector space V is a basis if it:

1. Spans V over F
2. B is linearly independent over F

Question 5

Ordered Basis of Vector Space

Answer 5

A basis of V where some extra information is provided: namely, which element of B comes “first”, which comes “second”, etc

Question 6

Coordinate Vector

Answer 6

A representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis

Question 7

Basis Theorem

Answer 7

If an F-vector space V has a

basis with n vectors, then every basis for V (as an F-vector space) has exactly n vectors.

Question 8

Finite and Infinitely Dimensional Vector

Answer 8

A vector space V vector spaces well defined is finite-dimensional if it has a basis
consisting of finitely many vectors. If not, it is infinitely dimensional