Is RA vector space over Q?

Is RA vector space over Q?

Is RA vector space over Q?

We’ve just noted that R as a vector space over Q contains a set of linearly independent vectors of size n + 1, for any positive integer n. Hence R cannot have finite dimension as a vector space over Q. That is, R has infinite dimension as a vector space over Q.

Which of the following is a Vectorspace over the field Q?

To see that Qn is a vector space over Q you could first prove that every field F is a vector space over itself and then since the product of vector spaces is a vector space Fn is a vector space over F and so Qn is a vector space over Q since Q is a field.

Is R2 a 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 an F vector space?

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 .

What is the basis of R over Q?

In fact, because Q is countable, one can show that the subspace of R generated by any countable subset of R must be countable. Because R itself is uncountable, no countable set can be a basis for R over Q.

What is the dimension of R over Q?

So, the existence of a finite basis for R as a vector space over Q would imply that R is countable. Thus, R is an infinite dimensional vector space over Q, leading to the conclusion that [R : Q] = ∞.

What is R 2 set?

(7) Let R2 denote the set of all ordered pairs of real numbers. That is, let R2 be the set which consists of all pairs (x, y) where x and y are both real numbers. We may think of R2 geometri- cally as the set of all points on the Cartesian coordinate plane.

What does R 2 vector mean?

Algebraically, a vector in 2 (real) dimensions is defined to be an ordered pair (x, y), where x and y are both real numbers (x, y ∈ R). The set of all 2 dimensional vectors is denoted R2.

What is the space F n?

The vector space Fn Then Fn forms a vector space under tuple addition and scalar multplication where scalars are elements of F. Fn is probably the most common vector space studied, especially when F=R and n≤3. For example, R2 is often depicted by a 2-dimensional plane and R3 by a 3-dimensional space.

Is RA vector space over R?

R is a vector space where vector addition is addition and where scalar multiplication is multiplication. (f + g)(s) = f(s) + g(s) and (cf)(s) = cf(s), s ∈ S.