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dschallipp
22.05.2020 •
Mathematics
Suppose h#; #i1 and h#; #i2 are inner products on v such that hv;wi1 d 0 if and only if hv;wi2 d 0. prove that there is a positive number c such that hv;wi1 d chv;wi2 for every v;w 2 v.
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Ответ:
We take v in the vector space V different from 0. Since v is not 0, then
Lets take k>0 such that
Now, we take any vector w. We want to show that
.
Since v is any non-zero vector, then this will prove that
for any vectors a,b. The reason is that for any vector different from 0, lets name it x, there will exist a constant
such that
for any y (this is for the same reason a constant exists for v). Since y can be anything, then it can be v. But that means that
, because v also has its constant k.
Now, lets show that
. Lets take a constant c such that
. We have that
Thus
Which means that
Which means that
. As a consequence
Which proves what we were looking for.
Ответ:
Step-by-step explanation:
Factor using the perfect square rule.
( 1 + 6 x ) 2