are the following vectors in R_2 resp. R_3 linearly dependent? Give reasons. (1) v1 = (2,1) and v2 = (1,-1)(2) u1 = (1,2,-1) and u2 = (5,1,4) and u3 = (6,1,5)  

Expert Answers
rcmath eNotes educator| Certified Educator

1) v1 and v2 are linear dependent if exist two non zero numbers a1 and a2 such that 

`a_1*v_1+a_2*v_2=0 =>`


`2a_1+a_2=0 and a_1-a_2=0=>` the second equation yield

`a_1=a_2 ` , substitue in the first yields that a1=a2=0.

Hence the vectors are linear independent.

2) Fof this example I will use a similar approach, but instead of adding them to zero, I will see if I can rewirte u3 in term of u1 and u2.




I will use the third one to check.

multiply the first equation by 2, then subtract you obtain



substitue in the first original you obtain



let's check our answers to see if we are going to obtain the 3rd coordinate of the vector.


Thus we can rewrite `u_3=-1/9*u_1+11/9*u_2`

Thus the vectors are linear dependent.