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.
Best Answer
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.