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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