1)Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R. Prove axioms (VS1)=For all x,y, x+y=y+x (commutativity of addition), (VS3)= There exist an element in V denoted by 0 such that x+0=x for each x in V.,(VS4)= For each element x in V there exist an element y in V such that x+y=0.,(VS5)= For each element x in V, 1x=x.,(VS7)=For each element a in F and each pair of elements x,y in V, a(x+y)= ax+ay. for this space.

2)Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.

Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

1)Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R. Prove axioms (VS1)=For all x,y, x+y=y+x (commutativity of addition), (VS3)= There exist an element in V denoted by 0 such that ...

Solution Summary

Vector spaces and scalar multiplication are investigated. The elements of functions in different spaces are determined.

... Definition: A vector space is a set of elements ... with two operations, addition and scalar multiplication, satisfying ... Let u, , and be vectors in , and let c and ...

... http://mathworld.wolfram.com/VectorSpace.html http://www.answers.com/topic/vector- space. ... Determine which sets are vector spaces under the given operations. ...

... Obviously, , and we have, for every + = and = Hence, is the ZERO matrix in the space. Matrices, Vector Spaces and Zero Vectors are investigated. ...

... where equality, addition and scalar multiplication are defined componentwise. Prove that V is a vector space over F . Modern Algebra Vector Spaces and Modules ...

... union of sub spaces. Vector Space A non-empty set of vectors V is called a vector space over a scalar field when the vector addition and scalar multiplication...

... is comprised of a detailed explanation of sub spaces of a vector space. ... an ,...) ai ∈ F , where equality, addition and scalar multiplication are defined ...

Linear Algebra - Vector Spaces. ... Proof: I want to show P is a vector space, I have to ... P <br>Axioms: Since we use usual addition and scalar multiplications, so we ...

... and let be the set of all sequences , , where equality, addition and scalar multiplication are defined ... Then is a vector space over . ... Notes:- Vector Spaces A non ...

Vecor Spaces and Linear Combinations. ... Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed ...