While there is a single additive identity (and it is NOT 0), each member of V has a different additive inverse. The definition of "additive identity" (I will call it "O" for the moment) is that "a+ O= a". Here, "addition" of vectors is defined as the product of the numbers: "a+ b= ab" above. Therefore, "a+ O"= aO= a. What number is O?jack_bauer said:Homework Statement
Define V =R with vector addition a+b=ab and scalar multiplication za=a^z.
Show that V is a vector space.
Homework Equations
a+b=ab, za=a^z
The Attempt at a Solution
I was able to check all the axioms but one, the additive inverse axiom where for all v in V there exists a... -v in V such that v+(-v)= 0. So far I have this: a+0=a(0)=0. In this case I believe 0 is the additive inverse.
A vector space is a mathematical structure that consists of a set of vectors and two operations: addition and scalar multiplication. These operations must satisfy certain properties, such as closure, commutativity, and associativity, in order for the set to be considered a vector space.
In order to show that a set is a vector space, you must prove that it satisfies all of the necessary properties. This includes showing that the set is closed under addition and scalar multiplication, and that the operations are commutative and associative. You must also prove the existence of a zero vector and the existence of additive and multiplicative inverses for each vector in the set.
A vector space is a set that satisfies all of the properties of a vector space, while a subspace is a subset of a vector space that also satisfies all of the properties. In other words, a subspace is a vector space within a larger vector space.
Linear independence is an important concept in vector spaces. A set of vectors is considered linearly independent if no vector in the set can be written as a linear combination of the other vectors. In other words, each vector in the set is necessary to span the entire vector space. Linear independence is a key property in determining whether a set of vectors forms a basis for a vector space.
Yes, a vector space can have multiple bases. A basis is a set of vectors that spans the entire vector space and is linearly independent. However, there can be more than one set of vectors that satisfy these criteria. For example, in a two-dimensional vector space, the vectors (1,0) and (0,1) form a basis, but so do the vectors (2,0) and (0,2). Both sets of vectors span the entire space and are linearly independent.