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hba

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

  • one year ago
  • one year ago

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  1. hba
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    @Hero

    • one year ago
  2. ParthKohli
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    I am overrated. I don't know all this stuff!

    • one year ago
  3. hba
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    No worries c:

    • one year ago
  4. hba
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    @phi

    • one year ago
  5. hba
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    I also do not know this stuff,I am just starting.

    • one year ago
  6. hba
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    I know we can use wronskian,determinant and definition. Can you explain me by definition ?

    • one year ago
  7. hba
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    @zaynahf

    • one year ago
  8. ghazi
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    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 :)

    • one year ago
  9. hba
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    Please explain further and in depth :)

    • one year ago
  10. ghazi
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    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

    • one year ago
  11. ghazi
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    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

    • one year ago
  12. hba
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    Thanks a lot for helping,I got it :)

    • one year ago
  13. ghazi
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    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

    • one year ago
  14. hba
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    @ghazi What :o Won't that be linearly independent ?

    • one year ago
  15. ghazi
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    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

    • one year ago
  16. hba
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    I am so confused Just answer me Linearly dependent or linearly indepdent ?

    • one year ago
  17. ghazi
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    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

    • one year ago
  18. hba
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    Thanks a lot dude got it :)

    • one year ago
  19. ghazi
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    you're welcome :D

    • one year ago
  20. ghazi
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    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

    • one year ago
  21. hba
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    What if there are only two vectors ?

    • one year ago
  22. ghazi
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    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

    • one year ago
  23. hba
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    So for my question would it be like, |dw:1358511169334:dw|

    • one year ago
  24. hba
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    @ghazi I know that but i am trying to learn all the methods.

    • one year ago
  25. hba
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    @ghazi Did i do this correctly ?

    • one year ago
  26. ghazi
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    do you think you can take determinant of the 2 x 3 elements that you have formed , i dont think so

    • one year ago
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