Are linearly dependent vectors always coplanar?

Question

If not, when would linear dependency imply coplanarity?

Answer 1

If we have three vectors that are linearly dependent, they are coplanar. This simply means that the third vector can be expressed as a linear combination of the other two.

 

If we have more than two coordinates, the above still holds. Three linearly dependent vectors are always coplanar.

 

The statement holds true, only the above happen.

 

If we have 4 vectors, and the three of them are linearly independent in 3D space, when a vector is a linear combination of all three vectors, then that vector is not coplanar with any of the base vectors in general case, when we need all three vectors to express it.