HallsofIvy said:tex looks good to me. Since you are doing this over the real numbers, yes, (1, 0) and (i, 0) work fine although I would see no reason to write the "0" and don't think you really need to write this as a pair. "1" and "i" as basis should do.
A real vector space is a mathematical structure that consists of a set of vectors and a set of operations, such as addition and scalar multiplication, that allow the vectors to be combined and manipulated. It follows a specific set of axioms and properties, such as closure under addition and scalar multiplication, to ensure that it behaves consistently.
Complex vectors are vectors whose components are complex numbers, rather than real numbers. They can be thought of as arrows in a complex plane, with both a real and imaginary component. Complex vectors are often used in mathematics and physics to represent quantities that have both magnitude and direction.
Complex vectors can be used in real vector spaces, meaning they follow the same axioms and properties as real vectors, as long as the operations are defined appropriately. For example, addition and scalar multiplication for complex vectors must follow the rules of complex arithmetic, but they still satisfy the properties of a real vector space.
The basis of a vector space is a set of vectors that can be used to represent any other vector in that space. It is the minimal set of vectors needed to span the entire space, meaning that any vector can be written as a linear combination of the basis vectors. The size of the basis is known as the dimension of the vector space.
A basis of a real vector space with complex vectors is defined in the same way as a basis of a real vector space with real vectors. It is a set of linearly independent vectors that span the entire space. However, the basis vectors themselves will be complex vectors, and the operations of addition and scalar multiplication will follow the rules of complex arithmetic.