If set contains more vectors then entries. the set islinearly dependent. Prove w

Question

If set contains more vectors then entries. the set islinearly dependent.

Prove with formal mathematical argument. This is intuitivelyobvious proof, making it difficult to formally prove.

Explanation / Answer

If set contains more vectors then entries. the set islinearly dependent. Prove with formal mathematical argument. This is intuitivelyobvious proof, making it difficult to formally prove. HOPE YOU MEAN THAT IN THE FIELD RM OF DIMENSION M THERE ARE N VECTORS WITH N>M LET THE VECTORS BE V1=[V11,V12,.......V1M]' V2=[V21,V22,........V2M]' ........................ VN=[VN1,VN2,.....VNM]' N>M TST THEY ARE L.D. -------------------------- WE CAN PROVE IN DIFFERENT WAYS. LET ME SHOW YOU ONE ARGUMENTATIVE PROOF LET US TAKE ANY M VECTORS V1,V2,.....VM FROM THE ABOVE. THERE ARE 2 POSSIBILITIES... CASE 1 THEY ARE L.D....THEN IT IS PROVED THAT THE SET OF NVECTORS OF DIMENSION M WITH N>M ARE L.D CASE 2 THEY ARE L.I. THEN WE HAVE A SET OF M LINEARLY INDEPENDENT VECTORS IN RM OF DIMENSION M. HENCE THEY CONSTITUTE A BASIS FOR RM... HENCE ANY OF THE N-M REMAINING VECTORS CAN BE EXPRESSED AS A LINEAR COMBINATION OF THESE M VECTORS. THAT IS THE SET OF N VECTORS OF DIMENSION M WITH N>M AREL.D. THAT IS IN ALL CASES THE SET OF N VECTORS OF DIMENSION M WITH N>M ARE L.D...............PROVED..

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