There is a definition in the book that says:
A set of n vectors {$e_1, e_2, …, e_n$} is linearly independent if there do not exist real numbers $a_1, a_2, …, a_n$, where at least one of the $a_i$ is not zero, such that $$a_1e_1+a_2e_2+…+a_ne_n=\vec0$$
Otherwise, the set {$e_1, e_2, …, e_n$} is called linearly dependent
I can't get into it at all. If we have a set of linearly independent vectors then we can't get $0$ vector after their linear combination, isn't it true?
There's also a theorem that states
Given two nonzero vectors $e_1$ and $e_2$, if $e_1 \cdot e_2 = 0$, then $e_1$ and $e_2$ are linearly independent
but that seems like it says about a narrow case where both vectors are orthogonal but vectors $v_1 = \begin{bmatrix}1, 0\end{bmatrix}$ and $v_2 = \begin{bmatrix}1, 1\end{bmatrix}$ are linearly independent because their span is a 2D plane, however, their dot product is not equal to $0$
I'm not a strong mathematician, the most simplistic explanations are preferred
Best Answer
The definition of linear independence that you gave is the classical one, but it doesn’t help much with intuition, in my view.
I think it’s better to start by defining linear dependence: a set of vectors is linearly dependent if one of them can be written as a linear combination of the other ones. That should make sense because it closely matches the meaning of “linearly dependent” in normal (non-mathematical) English. Then, of course a set of vectors is said to be linearly independent if they are not linearly dependent.
This definition is equivalent to the classical one you cited, but it makes a lot more sense (to me, anyway).
So, think of the three vectors $a = (1,0,0)$, $b=(0,1,0)$, and $c=(2,5,0)$ in $\mathbb R^3$. Obviously $c = 2a + 5b$. So these vectors are linearly dependent. It’s also true that $2a + 5b - c = 0$, so we have a (non-silly) linear combination that gives us zero, as required by the classical definition, but this is rather less intuitive, in my view.
In fact, in $\mathbb R^3$, three vectors are linearly dependent if and only if they are coplanar. This should make sense —- if $c$ lies in the same plane as $a$ and $b$, then $c$ can be written as a linear combination of $a$ and $b$. My example above is just a rather trivial example of three coplanar vectors. I expect you can invent more interesting ones.
If two vectors are orthogonal, then it’s pretty obvious that neither can be written as a multiple of the other, so they’re certainly linearly independent. But, as you point out, this is a very special case, and it’s easy to find examples of pairs of vectors that are linearly independent without being orthogonal.