any1 know ?
*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
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 R^{3}.
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).