Is R a vector space over C?

Is R a vector space over C?

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. EXAMPLE-2 R is a vector space over Q, because Q is a subfield of R.

Is C is not a vector space over R?

(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.

Is R is a vector space over R?

Since Rn = R{1,…,n}, it is a vector space by virtue of the previous Example. Example. R is a vector space where vector addition is addition and where scalar multiplication is multiplication. We call these operations pointwise addition and pointwise scalar multiplication, respectively.

What is a vector space over C?

Subtraction of two vectors and division by a (non-zero) scalar can be defined as. When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers C, the vector space is called a complex vector space.

Is r2 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 R NA vector space?

Definition and structures For any natural number n, the set Rn consists of all n-tuples of real numbers (R). With componentwise addition and scalar multiplication, it is a real vector space. Every n-dimensional real vector space is isomorphic to it.

Which of the following is not vector space over field R?

The following sets and associated operations are not vector spaces: (1) The set of n×n magic squares (with real entries) whose row, column, and two diagonal sums equal s≠0, with the usual matrix addition and scalar multiplication; (2) the set of all elements u of R3 such that ||u||=1, where ||⋅|| denotes the usual …

Is Raq a vector space?

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 . The answer is no.

Is R2 a known vector space?

The vector space R2 is represented by the usual xy plane. Each vector v in R2 has two components. The word “space” asks us to think of all those vectors—the whole plane. Each vector gives the x and y coordinates of a point in the plane : v D .

What is a vector space over R?

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 .