Question

Show that the following vectors are linearly independent.(Over C or R)

(1,1,1) and (0,1,-2)

Answer 1

@Hero

Answer 2

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

Answer 3

No worries c:

Answer 4

@phi

Answer 5

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

Answer 6

I know we can use wronskian,determinant and definition.

Can you explain me by definition ?

Answer 7

@zaynahf

Answer 8

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 :)

Answer 9

Please explain further and in depth :)

Answer 10

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

Answer 11

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

Answer 12

Thanks a lot for helping,I got it :)

Answer 13

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

Answer 14

@ghazi

What :o

Won't that be linearly independent ?

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

Thanks a lot dude got it :)

Answer 19

you're welcome :D

Answer 20

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 if the determinant is zero then vectors are linearly independent