Let s u, v be a linearly independent set. prove that the {u + v, u - v} is linearly independent

This means that there exist constant a,b such that if:

                                au+bv=0

                             then a=b=0

{u+v,u-v} is a linearly independent set.

Let us consider there exists constant c,d such that:

                            c(u+v)+d(u-v)=0

To show:   c=d=0

The expression could also be written as:

 cu+cv+du-dv=0

( Since, using the distributive property)

Now on combining the like terms that is the terms with same vectors.

cu+du+cv-dv=0

i.e.

(c+d)u+(c-d)v=0

Since, we are given that u and v are linearly independent vectors this means that:

c+d=0(1)

and c-d=0 i.e c=d(2)

and from equation (1) using equation (2) we have:

2c=0

i.e. c=0

and similarly by equation (2) we have:

         d=0

Hence, we are proved with the result.

We get that the vectors {u+v,u-v} is linearly independent.

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.