# Question about the associativity of Addition axiom for vectorspaces with respect to second degree polynomials

My question is concerning polynomials and the following axiom for real vector spaces

• Associativity of Addition: (Axiom 3) $$\displaystyle \mathbf{u} +(\mathbf{v} +\mathbf{w}) =(\mathbf{u} \ +\ \mathbf{v}) +\mathbf{w}$$

Below is problem 10, chapter 4.1 of Anton's book elementary linear algebra.

• Show that the set of polynomials of the form $$\displaystyle a_{0} +a_{1} x$$ with the operations below is a vector space. $$\begin{gather} ( a_{0} +a_{1} x) +( b_{0} +b_{1} x) =( a_{0} +b_{0}) +( a_{1} +b_{1}) x\\ k( a_{0} +a_{1} x) =( ka_{0}) +( ka_{1}) x \end{gather}$$

• Please note: I realize this is a subspace, so it is only enough to show that it is nonempty and closed under multiplication and addition. However, that's not what the question is asking for.

Below are two ways I show the axiom for associativity of addition.

Axiom 3: Solution 1 $$\begin{gather*} ( a_{0} +a_{1} x) +(( b_{0} +b_{1} x) +( c_{0} +c_{1} x)) \ =\ \\ ( a_{0} +( b_{0} +c_{0})) +( a_{1} +( b_{1} +c_{1})) x\ =\\ (( a_{0} +b_{0}) +c_{0}) +(( a_{1} +b_{1}) +c_{1}) x=\\ (( a_{0} +a_{1} x) \ +\ ( b_{0} +b_{1} x)) \ +\ ( c_{0} +c_{1} x) \ \end{gather*}$$ Axiom 3: Solution 2 $$\begin{gather*} ( a_{0} +a_{1} x) +(( b_{0} +b_{1} x) +( c_{0} +c_{1} x)) \ =\ \\ ( a_{0} +a_{1} x) +(( b_{0} +c_{0}) +( b_{1} +c_{1}) x) =\\ ( a_{0} +( b_{0} +c_{0})) +( a_{1} +( b_{1} +c_{1})) x\ =\\ (( a_{0} +b_{0}) +c_{0}) +(( a_{1} +b_{1}) +c_{1}) x=\\ (( a_{0} +b_{0}) +( a_{1} +b_{1}) x) +( c_{0} +c_{1} x) =\\ (( a_{0} +a_{1} x) \ +\ ( b_{0} +b_{1} x)) \ +\ ( c_{0} +c_{1} x) \ \end{gather*}$$

• My thoughts.
Solution 2 seems more correct because I feel like Solution 1 is imposing operations on the set V then expecting the vector space axioms to hold. However, they could both be wrong.

• Closing comments. The bulk of my confusion stems from how these proofs are demonstrated depending on the book I look at. Elementary Linear Algebra is a fantastic book, but the way they go about some of these proofs gives me a very loose understanding of the underpinning logic; my other book, linear algebra done right by axler, is verbose.

• So Solution $1$ is, basically, a sketch of Solution $2$, with some steps omitted. That is fine. A major part of writing proofs is not about being rigorous (we are always rigorous) but about communicating ideas. As such, a shorter proof, even if it skips some steps, may be more insightful, easier to read/teach, quicker to comprehend. Just as the centipede froze in horror when it started to think about which leg to move, too much detail may be counterproductive to understanding proofs. You need to tailor your proofs to your audience. Jan 14 at 18:58