Thread: Vector

*Choose two linearly independent vectors among the following vectors:
, ,


Extend them to a base in the room.*

Idk if this progress is wrong but here what i thought
(2,1,2)=y(0,0,2)+x(4,2,2) and i find that y = 1/2 and x = 1/2
idk what to do or if this is right thinking

any1 know ?

so on top it says Choose two linearly independent vectors among the following vectors:
and at the next text it says Extend them to a base in the room.
and svar means "answer"

let's pick 2 and see if there are linearly independent. i'm lazy, so i'll just pick the first 2.

now (4,2,2) and (0,0,2) are linearly independent if:

a(4,2,2) + b(0,0,2) = (0,0,0) forces us to pick a = b = 0.

now a(4,2,2) + b(0,0,2) = (4a,2a,2a) + (0,0,b) = (4a,2a,2a+b).

if (4a,2a,2a+b) = (0,0,0) we have:

4a = 0
2a = 0
2a+b = 0

from 4a = 0 (or 2a = 0), we see we must have a = 0. then:

2a+b = 0 becomes:

0+b = 0
b = 0, so we see (4,2,2) and (0,0,2) are, in fact, linearly independent.

presumably, you mean extend {(4,2,2),(0,0,2)} to a basis for all of R³.

to do this, we need a third vector NOT in span({(4,2,2),(0,0,2)}).

so we need a vector NOT of the form (4a,2a,2a+b).

can you think of one? (hint: pick a vector whose 2nd coordinate is not ____ of the first one).

hi deveno
im pretty new on vector and i keep read and search but dont get it :/ is it anyway i can draw this so i can find it more easy ?