Need help with Linear algebra thanks Prove that one cannot h

Need help with Linear algebra... thanks!

Prove that one cannot have a pair of two-dimensional subspaces, U, V R^2 with U V = {0} if U has linearly independent vectors u_1, u_2 and V has linearly independent vectors upsilon_1, upsilon_2. Then give the geometric interpretation of this.

Solution

Let U = span{ u₁ ,u₂} and V= span{v₁ , v₂} where u, u are linearly independent and v, v are also linearly independent and both U and V are subspaces of R². Since a standard basis for R² is { e₁ , e₂ }, hence both u₁, u₂ and also v₁ , v₂ have to be linear combinations of e₁ and e₂ . This implies that U = V = R². Hence U V = R² . Therefore, U V {0}.

A geometric interpretation of a 2 dimensional vector space in the Euclidean space is a plane passing through the origin. The origin itself forms the singleton {0} which is the only 0-dimensional subspace.