The set C(D,R) of all continuous real-valued functions defined over a given subset D of the real numbers is a real vector space: if x ↦→ f(x) and x ↦→ g(x) are continuous functions on D then so are x ↦→ f(x) + g(x) and x ↦→ cf(x) for all real numbers c; moreover these operations of addition of functions and of …

Vector: In medicine, a carrier of disease or of medication. For example, in malaria a mosquito is the vector that carries and transfers the infectious agent. In molecular biology, a vector may be a virus or a plasmid that carries a piece of foreign DNA to a host cell.

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

The set of real-valued even functions defined defined for all real numbers with the standard operations of addition and scalar multiplication of functions is a vector space.

Let V be the set R2 with the following operations defined as follows: for any (x1,y1), (x2,y2) ∈ R2, define (x1,y1)+(x2,y2) = (2(x1 + y1 + x2 + y2), −1(x1 + y1 + x2 + y2)). for any k ∈ R, and for any (x1,y1) ∈ R2, define k(x1,y1)=(kx1, ky1). Is V together with these operations a vector space?

(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.

The set FF is a vector space over F with pointwise operations: (f+g)(b):=f(b)+g(b),(af)(b):=af(b) for f,g∈FF and a,b∈F. Prove that the set UE of all even functions and the set UO of all odd functions are subspaces of FF. Bonus: Show that FF=UE⊕UO.

The constant function 0 is an odd function, and odd functions are closed under addition and scalar multiplication. Therefore the set of odd functions form a subspace of all functions.

A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections.

In other words, to test if a set is a subspace of a Vector Space, you only need to check if it closed under addition and scalar multiplication. Easy! ex. Test whether or not the plane 2x + 4y + 3z = 0 is a subspace of R3.

Every vector space, and hence, every subspace of a vector space, contains the zero vector (by definition), and every subspace therefore has at least one subspace: It is closed under vector addition (with itself), and it is closed under scalar multiplication: any scalar times the zero vector is the zero vector.