Question 22333: proofs kill me! S=(v1,v2,...vn) is a linearly independent set of vectors in the vector space V, prove that any nonempty subset of S must be linearly independent.
I know that all subsets are nonempty because they all contain the 0 vector which makes it nonempty, but I don't know how to prove that its linearly independent. Do i perhaps solve the matrix to equal zero, showing that it is independent?
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! S=(v1,v2,...vn) is a linearly independent set of vectors in the vector space V, prove that any nonempty subset of S must be linearly independent.
I know that all subsets are nonempty because they all contain the 0 vector which makes it nonempty, but I don't know how to prove that its linearly independent. Do i perhaps solve the matrix to equal zero, showing that it is independent?
WE CAN PROVE THIS BY REDUCTIO-AD-ABSURDUM ...THAT IS BY CONTRADICTION
WE ARE GIVEN THAT
S=(V1,V2,V3,.....Vn) IS SET OF LINEARLY INDEPENDENT VECTORS
THAT IS IF A1V1+A2V2+....AnVn =0 THEN A1=A2=...An=0
NOW WE HAVE TO prove that any nonempty subset of S must be linearly independent. LET US ASSUME IT WRONG .LET THERE BE ONE SUBSET
SAY T=(V1,V2,V3,....Ve)WHICH IS LINEARLY DEPENDENT
THAT MEANS WE SHALL HAVE
SAY....B1V1+B2V2+....+BeVe=0..........EQN.I
WITH ATLEAST ONE OF B1,B2,....Be....... NOT EQUAL TO ZERO.
NOW LET US ADD 0*Bf+0*Bg.........+0*Bn TO EQN.I
WE GET
B1V1+B2V2+...+BeVe+0*Vf+0*Vg+.....+0*Vn=0......EQN.II
WITH ATLEAST ONE OF B1,B2,....Be...... NOT EQUAL TO ZERO
THAT IS V1,V2,..........VnIS LINEARLY DEPENDENT...THIS IS A CONTRADICTION AS IT IS GIVEN THAT THEY ARE LINEARLY INDEPENDENT .HENCE OUR ASSUMPTION THAT THERE IS ONE SUBSET
SAY T=(V1,V2,V3,....Ve)WHICH IS LINEARLY DEPENDENT IS WRONG .....SO THERE IS NO SUBSET OF S WHICH IS LINEARLY DEPENDENT.HENCE ANY NON EMPTY SUBSET OF S IS LINEARLY INDEPENDENT.
THIS PRINCIPLE OF REDUCTIO-AD-ABSURDUM IS VERY IMPORTANT AND USEFULL IN MATHS .PLEASE UNDERSTAND IT AND PUT TO GOOD USE.
2.REDUCTIO-AD-ABSURDUM : Name looks scary and in fact many students to day do not know it by this name though they were taught and using it from 8/9th.class.This is how our Teacher explained….
Your Mother told “Don’t jump from a wall ! If you jump from a wall, you will break your bones”. This is a theorem to be proved. So let us prove it by assuming that the conclusion (to be proved ) is wrong. So jump from the wall.. you will break a bone…So our assumption is wrong….Hence the theorem is proved…
This is a method of proving a theorem by contradiction. We start by assuming that the conclusion (to be proved) is wrong. Then we end up with a contradiction to a known truth or the hypothesis (given data). Hence we conclude that our assumption is wrong….that is the theorem is correct.
|
|
|
