1 point Select the best statement. O A. If a set of vectors includes the zero ve

Question

1 point Select the best statement. O A. If a set of vectors includes the zero vector 0, then the set of vectors can span R as long as it contains n vectors. OB. If a set of vectors includes the zero vector 0, then there is no reasonable way to determine if the set of vectors spans R O C. If a set of vectors includes the zero vector 0, then the set of vectors cannot span R" O D. If a set of vectors includes the zero vector 0, then the set of vectors can span R" as long as the other vectors are distinct. O E. If a set of vectors includes the zero vector O, then the set of vectors spans R"precisely when the set with O excluded spans RT F. none of the above

Explanation / Answer

The answer C is true .

For prove our claim, we state a necessary and sufficient theorem :

Throrem: if V is a finite dimension vector space that is dim(V) =n, then set of n vectors of V spans V if and only if they are linearly independent.

Now we know that , R^n is vector space with dimension 'n'., So if it has n linearly independent set of vectors then they will span R^n.

But when a set of vectors contains a zero vector then the set of vectors are linearly dependent. So they can't be spanned R^n, by the above theorem. Hence option (c) is true other options are all false.

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