Question

Are these set of vectors linearly dependent or linearly independent?

(1,0,0,0) (1,1,0,0) (1,1,1,0) (1,1,1,1)

I say there are independent since I cannot think of any linear combinations to find the other vectors.

Answer 1

Can someone confirm my answer? If I am wrong can someone explain why?

Answer 2

definitely independant

Answer 3

theres an intuitive way to say it is

Answer 4

the more formula way is to take the determinant

Answer 5

so is this the case where I have to make an augmented matrix by multiplying c1(v1), c2(v2) or is that a different situation ?

Answer 6

what do u mean

Answer 7

oh um i see, okay so you mean the formal definition of linear independance

Answer 8

c1v1+c2v2+c3v3+c4v4=0
if c1 to c4 are constants and v1 to v4 are your vectors given
if the solution to this equation is such that all c1 to c4 are 0, the only trivial solution, then these vectors are said to be lienarly independant

Answer 9

Yes. So I can do that for the given problem above to prove that it is linearly independent by finding all solutions to be 0 ?

Answer 10

yeah thats hard to do, to check for all combinations

Answer 11

have you heard of determinants?

Answer 12

does your teacher not want u to use determinant

Answer 13

Yup we have covered determinants. How he did it in class was he made an augmented matrix with all 0's at the end (he called it homogenous) and then he solved for the solutions after doing REF

Answer 14

yea i was gonna go into that after the basic method

Answer 15

there are more intuitive ways to do this

Answer 16

one of them is
consider the vectors you are given, they are dimension 1,2,3,4, in order

Answer 17

so consider vectors 1 and 2, they haven o dimension 3, so they cant possible effect he 3rd vector in such a way that their can ever give u the 3rd vector

Answer 18

you can think of how to extend htis argument

Answer 19

now another method is like ur prof did, to rewrite your vectors

Answer 20

Yes that makes sense.

Answer 21

u can eliminate a vector and rewrite it as a combination of the other vectors, and that should still be linearly independant

Answer 22

if these 4 were lienarly independant to begin with

Answer 23

okay. So if u were to use the determinant method, does the determinant have to be 0 to indicate that the set of vectors are linearly independent?