Determining Vector Spaces: R^2 and M_2,2 Homework Solutions

[joemama69]

Homework Statement

Determine if the following sets are vector spaces

Part A) R^2; u+v = (u1 + v1, u2 + v2); cu = (cu1,cu2)

part b) M_2,2

A + B = |(a_11 + b_11) (a_12 + b_12)| & cA = |ca_11 ca_12|
|(a_21 + b_21) (a_22 + b_22)| |ca_21 ca_22|

sorry this is a little crude but those are both suppose to be 2X2 matricies

Homework Equations

The Attempt at a Solution

Part A) Yes, it is a vector space because it follows the addition & multiplication properties of vector spaces

Part B) yes it is a vector space because the set is closed under matrix addition & sclar multiplication


Are these statements true and a sufficient answer

 

[Wingeer]

If they both are closed under scalar multiplication and addition, then you can conclude that they might be a subset of a given vector space. To check whether or not they are vector spaces, you have to go through the 10 axioms. Does it sound familiar?

 

[VeeEight]

Your solution is not sufficient because you need to go through each and every vector space axiom to prove that it is in fact a vector space (and you need to show where only one axiom fails to prove it is not a vector space).

It is easy to see that pointwise addition is a simple procedure, but I think the point of the exercise is to go through the seven vector space axioms - many are trivial but as far as it stands now your answer is not sufficient.