Show that the vectors are linearly dependent by finding a vector in the list that is a linear combo of predecessors..

Question

anonymous · Answer

Specify the first vector in the list with this property.. v_1(1,1,1), v_2=(1,1,0) v_3=(0,0,5)

anonymous · Answer

i'm suppose to solve these with inspection but I can't see it..

anonymous · Answer

v_1 is a linear combination of v_2 and v_3.
Namely:
$$v_1=v_2+\frac{1}{5}v_3$$

anonymous · Answer

how you know it's 1/5v_3?

anonymous · Answer

Because when they say "linear combination" it can be any combination with any scalar multiple. So since v_3 is <0,0,5> and v_1 has a 1 in the 3rd "entry" you need to scale it by a factor of 1/5.

anonymous · Answer

sorry I don't see how v_1 is the linear combo of v_2 and v_3?

anonymous · Answer

I did the matrix and rref

anonymous · Answer

Okay, well take v_2 which is <1,1,0> and take (1/5) of v_3 which is <0,0,1> If you add them you get: <1,1,0>+<0,0,1>=<1+0,1+0,0+1>=<1,1,1>=v_1

anonymous · Answer

oh so you just have to find some way to add them to get back a vector

anonymous · Answer

Exactly, a linear combination :P Just don't let the words confuse you.

anonymous · Answer

okay thanks for your help make sense

anonymous · Answer

You're welcome :P If you have anymore just post them xD