u=

(5)

(1)

(2)

v=

(1)

(2)

(1)

w=

(3)

(1)

(-2)

is it u+v+w = 0 if they are linearly independent?

And how do i find out if its form a basis ofR^3

Printable View

- August 26th 2010, 06:47 PMjhtlinearly independent?
u=

(5)

(1)

(2)

v=

(1)

(2)

(1)

w=

(3)

(1)

(-2)

is it u+v+w = 0 if they are linearly independent?

And how do i find out if its form a basis of**R**^3 - August 26th 2010, 06:55 PMtonio

What you have to check is whether is possible with at least one of the scalars not zero. In our case you've to check whether:

.

If the above homogeneous linear system only has the trivial solution then the three vectors are lin. ind., otherwise they're lin. dep.

Tonio - August 26th 2010, 07:31 PMjht
Thanks, i also have a question other than part (b) above,

say if u,v,w is lin,indep

q=w-v+lu and l E**R**

show u,v,q is lin,indep.

I can see u,v,q is lin,indep if q=w-v+u

But what if the equaition at above with a l real number multiply by u in the equation? - August 26th 2010, 07:47 PMjht
so i found a=20/63 ; b = -5/9; c= 5/7

i am wondering is it right? - August 27th 2010, 09:31 AMHallsofIvy
No, if there exist

**any**numbers, a, b, c, not all 0, so that au+ bv+ cw= 0 then they are**dependent**if u+ v+ w= 0 then a= b= c= 1 are not all 0 so they are dependent. Of course, with the u, v, w you give here, u+ v+ w is NOT 0.

Quote:

And how do i find out if its form a basis of**R**^3

- August 27th 2010, 09:37 AMHallsofIvy
- August 27th 2010, 09:53 AMHallsofIvy
Suppose u, v, q were NOT linearly independent. The there exist a, b, c, not all 0, such that au+ bv+ cq= 0. But q= w- v+ lu so that is au+ bv+ c(w- v+ lu)= (a+ lc)u+ (b- c)v+ cw= 0. Since u, v, and w

**are**independent, we must have a+ lc= 0, b- c= 0, and c= 0. Those reduce to a= 0, b= 0, and c= 0 contradicting the assertion that they are not all 0. Since c= 0, the "l" in lc does not matter.