# Vector Space and Subspace

Define vectors pace and subspace with examples.

State and prove a necessary and sufficient condition for a subset of vectors to be a subspace.

Show that the intersection and union of two sub spaces are also sub spaces.

https://brainmass.com/math/linear-algebra/vector-space-subspace-625063

#### Solution Preview

The solution illustrates the detailed explanation of the following:

i) Definition of vectors pace and subspace with examples.

ii) A necessary and sufficient condition for a subset of vectors to be a subspace.

iii) Nature of the intersection and union of sub spaces.

Vector Space

A non-empty set of vectors V is called a vector space over a scalar field when the vector addition and scalar multiplication operations satisfy the following properties.

A1) for all

A2) for all

A3) for every

A4) There exist an element such that for all

A5) For each , there is an element such that

M1) for all and

M2) for all and

M3) for all and

M4) for all and

M5) ...

#### Solution Summary

The solution provides the definition and properties of vector space and subspace.