A basis is a set of linearly independent vectors that span a vector space. In other words, every vector in the vector space can be written as a linear combination of the basis vectors.
Proving that a set of vectors cannot be a basis is important because it helps us understand the limitations and properties of vector spaces. It also allows us to identify when a set of vectors is not suitable for certain mathematical operations.
A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the other vectors. In other words, no vector in the set is redundant or can be removed without changing the span of the set.
We can prove that a set of vectors cannot be a basis by showing that the vectors are linearly dependent or by demonstrating that they do not span the vector space. In this case, we can show that the vector u+v can be written as a linear combination of u and v, making the set linearly dependent.
Yes, there are alternative methods to prove that a set of vectors cannot be a basis. One method is to show that the vectors are not orthogonal, meaning they are not perpendicular to each other. Another method is to demonstrate that the vectors do not form a complete set, meaning they do not span the entire vector space.