Can coplanar vectors be linearly independent?

Note that any two vectors are coplanar, and any their linear combination is a vector lying in the same plane. If does not lie in the same plane, then it cannot be expressed as a linear combination of . Hence, a set of three non-coplanar vectors is linear independent.

Is Uvw linearly independent?

Since (u,v,w) are linearly independent, the only solution to this equation is (a+c)=(a+b)=(b+c)=0.

Is u v w z linearly dependent?

No, let x=v, is a redundant vector in {u,v,w,z}. So, {u,v,w,z} is linearly independent.

Are two perpendicular vectors linearly independent?

We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero. Definition. Proposition An orthogonal set of non-zero vectors is linearly independent.

Are coplanar vectors linearly dependent?

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.

Are the three vectors u v u w and v W also linearly independent?

So {u+v,u+w,v+w} is linearly independent.

What is linear independence of vectors?

In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be linearly independent. These concepts are central to the definition of dimension.

Are perpendicular lines linearly independent?

Every set which contains mutually perpendicular vectors is a independent set.

Can vectors be linearly independent but not orthogonal?

No! Two vectors are linearly dependent if and only if one is a scalar multiple of the other. For example, and are linearly independent, but , so they are not orthogonal.