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Homework Statement
Prove that the coordinates of a vector v in a vector space Vn are unique with respect to a given basis B={b1,b2,...,bn}
Homework Equations
The Attempt at a Solution
not sure at all what to do with this
It means that for a given basis, there is only one set of coordinate vectors that can represent a vector in a vector space. In other words, no two vectors can have the same set of coordinate vectors for a given basis.
To prove that coordinate vectors are unique, we need to show that for any vector in the vector space, there is only one set of coordinates that can represent it with respect to the given basis. This can be done using the properties of linear independence and spanning.
Yes, coordinate vectors can be unique for different bases. This is because the choice of basis determines the set of coordinate vectors that can represent a vector in a vector space. Different bases will have different sets of coordinate vectors that are unique to them.
Proving the uniqueness of coordinate vectors for a given basis is important because it ensures that we have a consistent and reliable way of representing vectors in a vector space. It also allows us to perform operations on vectors, such as addition and scalar multiplication, using their respective coordinate vectors.
In certain situations, coordinate vectors may not be unique for a given basis. This can occur when the vector space is non-Euclidean or when the basis is not orthogonal. In these cases, there may be multiple sets of coordinate vectors that can represent a vector in the vector space with respect to the given basis.