Understanding vector space axioms

Question

I am completely lost on the idea of vector spaces. I have read notes and watched videos and I am so confused. Can someone give me the general idea as to how I am supposed to figure out how these two satisfy the axioms of the vector spaces?

Let $V$ be the set of vectors in $\mathbb R^2$ with the following definition of addition and scaler multiplication:

• Addition: $$\begin{bmatrix}x_1\\x_2\end{bmatrix}\oplus\begin{bmatrix}y_1\\y_2\end{bmatrix}=\begin{bmatrix}0\\x_2+y_2\end{bmatrix}$$
• Scaler Multiplication: $$\alpha\odot\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}\alpha x_1\\\alpha x_2\end{bmatrix}$$ Determine which of the Vector Space Axioms are satisfied.

Answer 1

As $V$ must be a group under $\oplus$, there must exist a neutral element $\begin{bmatrix}n_1\\n_2\end{bmatrix}$ with the property ${\begin{bmatrix}x_1\\x_2\end{bmatrix}}\oplus\begin{bmatrix}n_1\\n_2\end{bmatrix}=\begin{bmatrix}x_1\\x_2\end{bmatrix}$ for all $\begin{bmatrix}x_2\\y_2\end{bmatrix}$. By comparing with the definition of $\oplus$ we conclude $x_1=0$ for all $x_1\in\mathbb R$, a contradiction.

Answer 2

A vector space is a mathematical structure used to model linear combinations. That is, when we want to analyse the consequences of expressions like $\lambda_1 x_1 + ... + \lambda_n x_n$ the minimal structure we need in a set is that of a vector space, so that the sums of what we call vectors $x_1+x_2+...$ are well defined as is the method of combining them with scalars $\lambda$.

The vectors can then be seen as arrows, points, polynomials or any other object, and the scalars are elements of a field, such as the rational numbers.

To prove something is a vector space you have to show you can add vectors and produce a new vector that is still in your set, you need to have an additive inverse for each x and you need to check multiplication by scalars satisfies the distributive properties. In the language of algebra, you say a vector space is a group under addition, and closed under scalar multiplication.