Currently I am studying a section from my book on vector spaces. I'm having issues in understanding how I am supposed to prove some of the questions in the Exercises section, such as:
In each of the following, determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails.
13. $M_{4,6}$ with the standard operations.
14. $M_{1,1}$ with the standard operations.
15. The set of all third-degree polynomials with the standard operations.
16. The set of all fifth-degree polynomials with the standard operations.
17. The set of all first-degree polynomial functions $ax+b$, $a\neq 0$, whose graphs pass through the origin with the standard operations.
18. The set of all quadratic functions whose graphs pass through the origin with the standard operations.
I don't know how exactly to identify which axiom fails. Here are the axioms:
- $\mathbf{u}+\mathbf{v}$ is in $V$. Closure under addition.
- $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$. Commutative property.
- $\mathbf{u}+(\mathbf{v}+\mathbf{w}) = (\mathbf{u}+\mathbf{v})+\mathbf{w}$. Associative property.
- $V$ has a zero vector $\mathbf{0}$ such that for every $\mathbf{u}\in V$, $\mathbf{u}+\mathbf{0}=\mathbf{u}$. Additive identity.
- For every $\mathbf{u}\in V$, there is a vector in $V$ denoted by $-\mathbf{u}$ such that $\mathbf{u}+(-\mathbf{u}) = \mathbf{0}$. Additive inverse.
- $c\mathbf{u}$ is in $V$. Closure under scalar multiplication.
- $c(\mathbf{u}+\mathbf{v}) = c\mathbf{u}+c\mathbf{v}$. Distributive property.
- $(c+d)\mathbf{u}=c\mathbf{u}+d\mathbf{u}$. Distributive property.
- $c(d\mathbf{u})= (cd)\mathbf{u}$. Associative property.
- $1(\mathbf{u}) =\mathbf{u}$. Scalar identity.
Say we look at question 17: In the answer book it says that axiom 4 fails, but I don't see how that is possible.
Say if you have 4x+1, by axiom 4 you are supposed to add 0 to the vector u, so you would get 4x+1+0=4x+1, which is true....
I'm really not sure how to get my head around these type of problems. Can someone give me a coherent explanation? Am I supposed to test by hand each axiom, or just in my head?