Is C is a vector space over C?

No is not a vector space over . One of the tests is whether you can multiply every element of by any scalar (element of in your question, because you said “over ” ) and always get an element of .

Is a vector space over the field of complex numbers?

(i) Yes, C is a vector space over R. Since every complex number is uniquely expressible in the form a + bi with a, b ∈ R we see that (1, i) is a basis for C over R. Thus the dimension is two. (ii) Every field is always a 1-dimensional vector space over itself.

Is R NA vector space over C?

a vector space over its over field. For example, R is not a vector space over C, because multiplication of a real number and a complex number is not necessarily a real number. respect to the addition of matrices as vector addition and multiplication of a matrix by a scalar as scalar multiplication.

Is a vector space over?

A vector space over F — a.k.a. an F-space — is a set (often denoted V ) which has a binary operation +V (vector addition) defined on it, and an operation ·F,V (scalar multiplication) defined from F × V to V . (So for any v, w ∈ V , v +V w is in V , and for any α ∈ F and v ∈ V α·F,V v ∈ V .

Is R is a vector space over R?

Yes. Indeed the set of real nos. R is a Vector-space over the set of rationals Q . Because every field can be regarded as a Vector- space over itself or a sub – field of itself.

What does vector space over R mean?

A vector space over R is a nonempty set V of objects, called vectors, on which are defined two operations, called addition + and multiplication by scalars · , satisfying the following properties: A1 (Closure of addition) For all u, v ∈ V,u + v is defined and u + v ∈ V .

What is the basis of C over C?

Consider the vector space C over C. Then {1} is the standard basis, and the dimension is one. Every complex number can be written as a C-multiple of 1, so {1} is a spanning set for the space.

What is the dimension of the complex vector space?

If V is a complex vector space, we can consider only multiplication of vectors by real numbers, thus obtaining a real vector space, which is denoted VR. is a basis in VR, so the dimension of VR is 2n. They usually express this as dimRV = 2n, keeping the same symbol for V and VR.