hba 2 years ago Show that the following vectors are linearly independent.(Over C or R) (1,1,1) and (0,1,-2)

1. hba

@Hero

2. ParthKohli

I am overrated. I don't know all this stuff!

3. hba

No worries c:

4. hba

@phi

5. hba

I also do not know this stuff,I am just starting.

6. hba

I know we can use wronskian,determinant and definition. Can you explain me by definition ?

7. hba

@zaynahf

8. ghazi

well for these two vectors to be linearly DEPENDENT, vector 1 should be able to be expressed as product of a constant and the another vector which means \[V _{1}=KV _{2}\] is this helpful ? or shall i explain further :)

9. hba

Please explain further and in depth :)

10. ghazi

lets say \[V _{1}= i+ j+k \] and vector \[V _{2}=j-2k\] now we can see that \[V _{1} \neq KV _{2}\] therefore these are linearly independent vectors

11. ghazi

if we had two vectors like \[V _{1}= i+j+k\] and \[V _{2}= 2i +2j+ 2k\] we can easily see that \[V _{1}=k V _{2}\] therefore these two are linearly dependent vectors , wherever this condition fails , it means vectors are independent

12. hba

Thanks a lot for helping,I got it :)

13. ghazi

you're welcome but do look for three vectors too , it uses determinant technique , in the above example we can see that V1=2 V2 , so its dependent

14. hba

@ghazi What :o Won't that be linearly independent ?

15. ghazi

if \[V _{1}= K V_{2}\] then its dependent because you can see that one can be expressed in terms of other but if this condition fails then both the vectors are independent

16. hba

I am so confused Just answer me Linearly dependent or linearly indepdent ?

17. ghazi

look if \[V _{1}= K V_{2}\] now , if \[V _{1}=i+j+k\] and \[V _{2}= 2i+2j+2k\] then we can see that \[V _{1}=2 *V _{2}\] therefore these vectors are LINEARLY DEPENDENT BUT IF THIS CONDITION FAILS AND ONE VECTOR IS NOT EXPRESSED BY THE HELP OF CONSTANT TO THE ANOTHER ONE THEN ITS INDEPENDENT WHICH MEANS \[V _{1} \ne K V _{2}\] as the two vectors given in your question :D

18. hba

Thanks a lot dude got it :)

19. ghazi

you're welcome :D

20. ghazi

if you have three vectors like \[V _{1}=i+2j+3k\]\[V _{2}= 2i+j+3k\]\[V _{3}= i+j+k\] then you have to take determinant of these p elements that is the coefficients of i, j, k in all the three vectors |dw:1358510982059:dw| if the determinant is zero then vectors are linearly independent

21. hba

What if there are only two vectors ?

22. ghazi

then form a determinant of 2 x 2 and check if its mod is zero or not but the method that i told you earlier is easier for two vectors

23. hba

So for my question would it be like, |dw:1358511169334:dw|

24. hba

@ghazi I know that but i am trying to learn all the methods.

25. hba

@ghazi Did i do this correctly ?

26. ghazi

do you think you can take determinant of the 2 x 3 elements that you have formed , i dont think so