LQ5 R1 PLEASE ANSWER PART B This is the second time posting

L.Q5 R1. PLEASE ANSWER PART B. This is the second time posting this question and part B was not answered. ANSWER PART B. Thank you

Determine whether the vectors w=(1,1,3) =(-3,0,4) form a linearly dependent set or a linearly independent set. v=(5,-1,2) Consider the set of all triples of the form (a,b,c), where d\' + b2 =c2 . Check whether the set is a subspace of R3. b.

Solution

a.

Let, r,s,t so that

ru+sv+tw=0

r(-3,0,4)+s(5,-1,2)+t(1,1,3)=0

-3r+5s+t=0

-s+t=0 ie s=t

4r+2s+3t=0

Substituting s=t in first and third equations gives

-3r+6t=0 ie r=2t

4r+5t=0 ie r=-5t/4

Hence, r=s=t=0

Hence set is a linearly independent set

b.

No it is not a subspace.

Counterexample

(3,4,5) is one such triple

WE can rewrite it as

(4,3,5) which is also a triple

Adding the two gives

(3,4,5)+(4,3,5)=(7,7,10)

7^2+7^2=98 is not equal to 10^2

Hence not a triple

Hence sum of two elements in set is not in the set. Hence not a triple