- If I have 3 vectors $\vec{v_1}, \vec{v_2}$ and $\vec{v_3},$ can they be linearly independent in $\mathbb{R^4}$?
- If I have 5 vectors $\vec{v_1}, \vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5},$ can they be linearly independent in $\mathbb{R^4}$?
What is generally said about the linear dependency of $k$ number of vectors in $n$ number of dimensions?
I formulated the two statements below by myself. Is the reasoning in them correct?
- I know that for the span of the vectors, it follows that if $n<k,$
then the vectors $\vec{v_1},…,\vec{v_k}$ can't span
$\mathbb{R}^n$. For example 2 vectors can't span a space
$\mathbb{R}^3$. - If we have 4 vectors in $\mathbb{R^3}$ then that vector matrix would still have rank 3
and can still span $\mathbb{R^3}$.
Best Answer
Three vectors can be linearly independent in $\Bbb R^4$. Example: $v_1=(1, 0, 0, 0), v_2=(0, 1, 0, 0),$ and $v_3=(0, 0, 1, 0)$ are linearly independent vectors in $\Bbb R^4.$
You cannot have $5$ linearly independent vectors in $\Bbb R^4$. This is because $4$ vectors are necessary to span $\Bbb R^4$, and any additional vector will be a linear combination of those.
Statement one is correct if you switch $n$ and $k$ so that $k<n$. The example you give has $n=3$ and $k=2$. Aside from this, it is correct.
Statement two is correct.