Mr Davis 97 said:I have a simple question. Say we have some subspace that is nonempty and closed under scalar multiplication and vector addition. How could we deduce that ##0 \vec{u} = \vec{0}##?
A nonempty subspace is a subset of a vector space that contains at least one vector and satisfies the subspace properties, including closed under addition and scalar multiplication.
Proving 0u = 0 for a nonempty subspace means showing that the zero vector, or the vector containing all zeros, when multiplied by any vector in the subspace, results in the zero vector. This is an important property in linear algebra and is used in various mathematical proofs.
Proving 0u = 0 for a nonempty subspace is important because it helps establish the subspace as a true vector space. It also helps in proving other properties and theorems related to vector spaces.
Some techniques for proving 0u = 0 for a nonempty subspace include using the properties of vector addition and scalar multiplication, using mathematical induction, and using direct proof or contradiction. It is also important to understand the definitions and properties of vector spaces.
Yes, 0u = 0 can hold true for a nonempty subspace if the zero vector is in the subspace and the subspace satisfies the subspace properties, including closure under addition and scalar multiplication.