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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.
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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.

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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.

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  • $\begingroup$ Can you emphasize why the distributive property is failing to work? $\endgroup$ – KnowledgeGeek Feb 10 '14 at 19:34
  • $\begingroup$ That was a typo, it is the associative property that doesn't work. You can find three vectors such that (x+y)+z is not equal to x+(y+z) $\endgroup$ – Felipe Jacob Feb 10 '14 at 19:35
  • $\begingroup$ Is this the only axiom that fails to hold? $\endgroup$ – KnowledgeGeek Feb 10 '14 at 19:43
  • $\begingroup$ Is it clearer now? $\endgroup$ – Felipe Jacob Feb 10 '14 at 19:44
  • $\begingroup$ I guess I just don't know where to look or what to read to understand this topic. $\endgroup$ – KnowledgeGeek Feb 10 '14 at 19:44

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