The vector space axioms are a set of properties that define the rules for vector operations and their properties. These axioms include properties such as closure, commutativity, associativity, distributivity, and the existence of an identity element.
The vector space axioms provide a framework for proving that two vectors, v and w, are equal. By using these axioms, we can show that all the properties of v are also true for w, and vice versa, which ultimately proves that v and w are equal.
Yes, v=w can also be proven using other methods such as algebra or geometry. However, the use of vector space axioms is a more general and rigorous approach to proving vector equality.
One example is using the distributivity axiom to show that scalar multiplication of a vector is associative, and then using this to prove that v=0 if and only if w=0. Another example is using the closure axiom to show that if v=w, then v+w=w+v.
While the vector space axioms provide a powerful tool for proving vector equality, they do have limitations. Some vector spaces may not follow all of the axioms, making it impossible to use this method. Additionally, the proof may become complex for larger vector spaces.