Two linearly independent vectors in a plane that don't span the plane

[johnqwertyful]

Homework Statement

Say we have the plane, x+2y+4z=8 (part of a larger problem)

Homework Equations

The Attempt at a Solution

The vectors (8,0,0) and (0,0,2) both lie in the plane. They are linearly independent. But (0,4,0) lies in the plane and is not a linear combination of the first two vectors. How can this be? We have two linearly independent vectors in a two dimensional vector space that DON'T span it?

 

[Ray Vickson]

Homework Statement

Say we have the plane, x+2y+4z=8 (part of a larger problem)

Homework Equations

The Attempt at a Solution

The vectors (8,0,0) and (0,0,2) both lie in the plane. They are linearly independent. But (0,4,0) lies in the plane and is not a linear combination of the first two vectors. How can this be? We have two linearly independent vectors in a two dimensional vector space that DON'T span it?

The plane you have is not a subspace. A subspace would need to pass through the origin, so would need to have '0' on the right, not your '8'. Since you do not have a subspace, there is no reason to have the spanning property you want. Just draw a picture to see what is happening.

 

[johnqwertyful]

(0,0,0) is not in the plane, so this is not a vector space. Nevermind.

 

[johnqwertyful]

The plane you have is not a subspace. A subspace would need to pass through the origin, so would need to have '0' on the right, not your '8'. Since you do not have a subspace, there is no reason to have the spanning property you want. Just draw a picture to see what is happening.

Figured it out just after I posted, thanks though!

 

[LCKurtz]

Homework Statement

Say we have the plane, x+2y+4z=8 (part of a larger problem)

Homework Equations

The Attempt at a Solution

The vectors (8,0,0) and (0,0,2) both lie in the plane. They are linearly independent. But (0,4,0) lies in the plane and is not a linear combination of the first two vectors. How can this be? We have two linearly independent vectors in a two dimensional vector space that DON'T span it?

None of your three vectors lie in the plane. They are points in the plane. You can also think of them as position vectors to those points, which is why they aren't in the plane.