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prove $A$ is a subspace of $R^{2}$ with dimension 2

let $\lambda_1, \lambda_2 \in R$, a subspace of $\mathbb{R}^2$ with dimension 2 such that
$\begin{cases}A = \{\lambda_1 x + \lambda_2 y \mid x,y \in \mathbb{R}\\ \lambda_1,\lambda_2 \in R\}\end{cases}$
My main problem is with showing that this $A$ is a subspace of $\mathbb{R}^2$ with dimension 2.
Since $dim(A) = dim(\mathbb{R}^2)$ then we can say that $dim(A)$ must be at least 2 by dimension formula (possibly saying that $0\in A$). I know that $dim(A)$ must be 2 since this is the dimension of $R^2$. But the issue I am having is with showing that $A$ is a subspace of $\mathbb{R}^2$.
So, I know that $x_1\cdot x_2 = 0 \implies x_1\in A$ and $x_2\in A$ since $A$ is a subspace of $\mathbb{R}^2$. Then I know that the problem is with $y_1\cdot y_2$. It seems like from my previous proof that there is a problem here.
I know that

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Why not share? Scaryish legal music:

A posting code and a reference link to a government printed book is two-fold, for by that means you can prove to the next generation that the new faith or system came from the existing one, or the other way around, you can convince those who yet believe in the power of the existing system that there was once a time when they were wrong about it. The writing of the new faith and the teaching of the new system is first done on paper, book, gramophone record or tape or whatever. Only much later is the faith or system ‘officially’ put into action or a new system or system is introduced.

The book is the bible, the self-proclaimed authority says what it says, it has the ‘certification’ of the existing system so people are likely to trust it even if they do not believe it, and if other people praise the existing system it is then more accepted.

When next you look around and see how things have changed for the better, or for the worse, you will believe even less what others say, especially since they claim to understand the whole of the world and all that goes on in it. They simply will not be believed, they simply will not be understood. Any number of books, articles, etc, can be written by any person, the same way the same book could be written by different persons from two different systems, even by the same persons at different times. Books are the new gods, and can not be trusted because they are written by humans, who are really in it for themselves, who can not be believed.

So for every piece of knowledge you get, no matter how true or how false, it is always a human’s desire to gain money and power, and no less a desire to be famous. It would be impossible to believe anything a human says that does not paint its own picture of reality, this being so it is even impossible to believe a human who says there is only one true way of life and denies the existence of other ways.

I believe that the world is a place full of mysteries, and I will never understand why things are as they are.

So, to conclude:

You must never ask a human to show you more than you already believe, nor must you ever try to change its opinion of