Suppose I have two linearly independent vectors $u$ and $v$ in $\mathbb{R}^3$. Taking the cross product $u \times v$ gives me another linearly independent vector, which we call $w$. Then the collection $\{u, v, w\}$ forms a basis of $\mathbb{R}^3$. What seems odd to me is that although the dimension of $\mathbb{R}^3$ is $3$, we really "only need" two linearly independent vectors of $\mathbb{R}^3$ since we can just take the cross product of the two vectors, and so it feels like the dimension of $\mathbb{R}^3$ is actually 2. I guess my question then is, can someone help clarify this issue for me?
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