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magicarpet512
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If you find the exact same basis for two vector spaces, then is it true that the vector spaces are equal to each other?
A vector space is a mathematical structure that consists of a set of objects called vectors, which can be added together and multiplied by scalars (usually real numbers), and must satisfy certain axioms such as closure under addition and scalar multiplication.
The basis of a vector space is a set of linearly independent vectors that span the entire vector space. This means that any vector in the space can be expressed as a linear combination of the basis vectors.
The basis of a vector space allows us to represent any vector in the space using a unique set of coordinates, making it easier to visualize and manipulate. It also helps us to understand the structure and properties of the vector space.
Yes, a vector space can have multiple bases. In fact, any vector space with dimension greater than 1 will have infinitely many possible bases.
The basis of a vector space can vary depending on the field over which the vector space is defined. This means that the set of coefficients used to express a vector in terms of the basis vectors may be different. Additionally, the operations of addition and scalar multiplication may also be defined differently in different fields, leading to different properties and behaviors of the vector space.