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hba
Show that the following vectors are linearly independent.(Over C or R) (1,1,1) and (0,1,-2)
I am overrated. I don't know all this stuff!
I also do not know this stuff,I am just starting.
I know we can use wronskian,determinant and definition. Can you explain me by definition ?
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 :)
Please explain further and in depth :)
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
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
Thanks a lot for helping,I got it :)
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
@ghazi What :o Won't that be linearly independent ?
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
I am so confused Just answer me Linearly dependent or linearly indepdent ?
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
Thanks a lot dude got it :)
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
What if there are only two vectors ?
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
So for my question would it be like, |dw:1358511169334:dw|
@ghazi I know that but i am trying to learn all the methods.
@ghazi Did i do this correctly ?
do you think you can take determinant of the 2 x 3 elements that you have formed , i dont think so