Take 10 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
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Question
1. Prove that the familly of vectors of
is a basis of
2. Prove that the familly of vectors of
is linearly independent.
Use the dimensions of vector spaces
and
a) We saw in a previous exercise that the family of vectors of
s linearly independent, and we know that
has a dimension of
therefore the family of vectors of
is a basis of
b) Since we have a family of two vectors of
which has a dimension of
if the family is linearly independent it is a basis and therefore is also a generating family.
Let us consider a pair of real numbers
such that:
We infer :
We infer from the first equation
and by reporting this in the second one, we obtain
This implies
The family is linearly independent and with the previous reasonning it is a basis, hene a generating family.
