
Vector Spaces
Hi Im having trouble finding how to start some of these questions
For the definition of Vector Space over Ration Numbers I know that it is a set of v objects known as vectors, involved with additional operations to associate a pair of vectors and by using scalar multiplication operations will produce the product of rv vectors. Stating the multiplication and Additional laws.
However, 2 questions puzzled me such as:
1. (Real Numbers)R^3 be set of all triples of R. Ket S be subset of R^3 consisting triples x,y and z where z = 0. Prove S is a subspace of R^3. Let T be subset of R^3 consisting all triples of (x,y and z) where z=5. Would T be subset of R^3?
2. Let V be the set of all functions fL R > R so that
d^2f/dx^2 =f(x)
Prove that V is a subspace of Rvector space F(R,R) of all functions R>R where addition is defined by (f+g)(x)=f(x)+g(x) and ([gamma]f)(x) = [gamma](f(x)) for all x belonging to R.
Any tips would be well appreciated
Thx a lot