Need some math help thanks in advance!
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Answer:
No, those vectors do not span P₂
Step-by-step explanation:
At first, notice we have 4 vectors from a vector space of dimension 3 ({1,x,x²} is the standard base of P₂) so we are sure they are linearly dependent (because there are more than 3 vectors) i.e. it´s possible to write one of them in terms of other in the same set:
p4=p1-p2
Now, we conclude the span {p1,p2,p3,p4} = {p1,p2,p3}
If we want to know the subspace that this span gets, we work with the matrix of the vectors made as follows:
The row-reduction of this matrix show us the basis of the same row-subspace even in the case that some of them are linearly dependent:
This shows that the span {p1,p2,p3}={(1+0.5x²),(x-1.5x²)}
Looking in the previous equality, the span {p1,p2,p3} is generated by two linearly independent vectors, but P₂ needs 3 linearly independent vectors to be generated.
The answer is that {p1,p2,p3,p4}⊂P₂.