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