 # I keep hearing that if you have an RR^3 space you need 3 elements to fill the space. So, let's consider a column vector. I understood 3 elements to mean = 3 column vectors with 3 elements each. As you can't span RR^3 with JUST one or two vectors even if they each have 3 elements. Correct? Could you technically somehow take 2 column vectors with 4 elements to span RR^3? I'm guessing not. 2k1ablakrh0 2022-09-17 Answered
I keep hearing that if you have an ${\mathbb{R}}^{3}$ space you need 3 elements to fill the space.
So, let's consider a column vector. I understood 3 elements to mean = 3 column vectors with 3 elements each. As you can't span ${\mathbb{R}}^{3}$ with JUST one or two vectors even if they each have 3 elements. Correct?
Could you technically somehow take 2 column vectors with 4 elements to span ${\mathbb{R}}^{3}$? I'm guessing not.
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To the first question. You need three vectors ${v}_{1},{v}_{2},{v}_{3}\in {\mathbb{R}}^{3}$. It's correct. The elements refers to vectors $v\in {\mathbb{R}}^{3}$. But that is not enough to span ${\mathbb{R}}^{3}$, they have to be linearly independent; i.e.

The second question. No. First because a vector with 4 entries lives in ${\mathbb{R}}^{4}$. But if you consider ${\mathbb{R}}^{3}$ embedded in ${\mathbb{R}}^{4}$. Again you need 3 vectors ${v}_{1},{v}_{2},{v}_{3}\in {\mathbb{R}}^{4}$ that are linearly independent to span ${\mathbb{R}}^{3}$, but here the question is a little bit different. Think of $\mathbb{R}$ embedded in ${\mathbb{R}}^{2}$. You can embed it as any line that passes through the origin, so you have many representations of $\mathbb{R}$ in ${\mathbb{R}}^{2}$, same thing happens with ${\mathbb{R}}^{3}$ embedded in ${\mathbb{R}}^{4}$

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