If V n is the real vector space of all n-tuples ofreal numbers for each n>1, whi

Question

If Vn is the real vector space of all n-tuples ofreal numbers for each n>1, which of the following must betrue? 1.) Every basis of Vn contains exactly nvectors. 2.) Every basis of Vn is an orthogonal set ofvectors. 3.) Every set of n+1 vectors of Vn is a linearlydependent set. Please explain choices and why they are true. Thanks If Vn is the real vector space of all n-tuples ofreal numbers for each n>1, which of the following must betrue? 1.) Every basis of Vn contains exactly nvectors. 2.) Every basis of Vn is an orthogonal set ofvectors. 3.) Every set of n+1 vectors of Vn is a linearlydependent set. Please explain choices and why they are true. Thanks

Explanation / Answer

QuestionDetails: If Vn is the real vector space of all n-tuples ofreal numbers for each n>1, which of the following must betrue? 1.) Every basis of Vn contains exactly nvectors......................TRUE
THIS BY STANDARD THEOREM THAT N DIMENSIONAL VECTOR SPACE NEEDS NLINEARLY INDEPENDENT VECTORS TO CONSTITUTE THE BASIS
2.) Every basis of Vn is an orthogonal set ofvectors...................NOT NECESSARY
THE BASIS NEED NOT BE ORTHOGONAL
3.) Every set of n+1 vectors of Vn is a linearlydependent set.....................TRUE
AGAIN FOLLOWS FROM 1. ONE OF THE VECTORS IN THE N+1 SET OF VECTORSCAN BE EXPRESSED AS A LINEAR COMBINATION OF THE OTHER N VECTORS ,AS IF THE OTHER N VECTORS ARE NOT LINEARLY DEPENDENT THEYCONSTITUTE THE BASIS AND HENCE SPAN THE ENTIRE VECTORSPACE.
Please explain choices and why they are true. Thanks

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