Linear Subspace

DEFINITION1: Let K be a field, X be a K-vector space and M\subset X. If M is also a K-vector space, then M is called a linear subspace or vector subspace (or shortly subspace) of X.

PROPOSITION1: Let K be a field, X be a K-vector space and M\subset X. M is a subspace of X if and only if

a) \theta\in M,

b) x+y\in{M} for all x,y\in{M},

c) \lambda{x}\in{M} for all {\lambda}\in{K} and {x}\in{M}.

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Vector Space

DEFINITION1: Let K be a set with at least two distinct elements, +:K\times{K}\rightarrow{K} and \cdot:K\times{K}\rightarrow{K} be two functions. If the following conditions hold, then \left( K,+,\cdot \right) is called a field:

F1) \left( a+b \right) +c=a+ \left( b+c \right) for all a,b,c\in{K},

F2) a+b=b+a for all a,b\in{K},

F3) there exists {0}\in{K} such that a+0=a for all a\in{K},

F4) for each a\in{K}, there exists b\in{K} such that a+b=0,

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