A subset u1,...,up in V is linearly independent if and only if the set of...

djeljenike

djeljenike

Answered

2021-02-15

A subset u1,...,up in V is linearly independent if and only if the set of coordinate vectors
[u1]β,...,[up]β
is linearly independent in Rn

Answer & Explanation

l1koV

l1koV

Expert

2021-02-16Added 100 answers

A zero vector in V can be written as, c1u1+...+cpup=0.
Since the mapping x|[x]β is one-to-one,
the zero vector in Rn is,
[c1u1+...+cpup]β=[0]β
That is, [c1u1]β+...+[cpup]β=[0]β
Trivial solution of above equation implies that the equation
c1u1+...+cpup=0 also has trivial solution as the mapping is one-to-one and onto.
The trivial solution of [c1u1]β+...+[cpup]β=[0]β implies that
[c1u1]β,...,[cpup]β are linearly independent.
Only a trivial solution of c1u1+...+cpup=0 implies that vectors
u1,...,up are linearly independent.
Hence, a subset u1,...,up is linearly independent if and only
if the set of coordinate vectos [u1]β,...,[up]β is linearly
independent in Rn

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