How are vector spaces viewed as universal algebras?

Determine if these are vector spaces?

• I have NO IDEA on how to solve this. Exams tomorrow... please help T_T Determine if these are vector spaces. If it is not a vector space, explain what axiom that fail to hold. 1. The set of all 2x2 invertible matrices with the vector addition defined to be matrix multiplication: i.e., A + B = AB, and standard scalar multiplicaton. 2. The set of all real-valued functions f defined everywhere on the real line and such that f(1)=0, with the standard functional addition and scalar multiplication. 3. The set of all positive real numbers with the operations x+y = xy and kx=x^k 4. The set of all rational numbers, with the usual addition and scalar multiplication. HELPPPP

• Answer:

  Try this page: http://tutorial.math.lamar.edu/Classes/LinAlg/VectorSpaces.aspx I assume your underlying field is the real numbers, where we pull the scalars. 1. Is not a vector space since AB is not equal to BA in general, hence A + B is not equal to B + A. A vector space has to have a commutative addition operation. The rest of 2-4 I think are vector spaces and you can prove these by running through the list of axioms. Actually #3 has been worked out for you on the web link I posted at the beginning. #2 I recall from the days in graduate school doing the long abstract algebra homework sets; it's a famous example which carries over into fields and ideal theory.