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Question
Let
be a
-vector space.
1. Prove that a familly of vectors which contains a linearly dependent familly is linearly dependent.
2. Prove that a familly of vectors included into a linarly independent familly is linearly independent.
Since the subset is generating, there exists a zero linear combination with at least one non-zero coefficient. Complete this linear combination into a linear combination of the set with the missing vectors affected with zero coefficients.
a) Let
be a non linearly independent family.
Hence there exists a family
of real numbers that are not all zero, such that:
We consider a family
and we define a family of real numbers
by setting
is a family of real numbers that are non all zero and
hence the family of vectors
is linearly dependent.
b) Let
be a linearly independent family.
For a natural integer
we consider the family
and a linear combination of the family equal to the zero vector, therefore,
By setting again
for any
between
and
we obtain a linear combination of the vectors of the family equal to the zero vector:
The family of vectors
is linearly independent, therefore all the coefficients of the zero linear combination are zero, especially
hence the family
is linearly independent.
