Let $\operatorname{span}\{v_1, v_2, v_3\}$ be a basis for $\mathbb{R}^3$. This means that these vectors are linearly independent.
Thus, $a v_1 + b v_2 + c v_3 = 0$. However how can it make sense that the addition of three vectors give a real value. If you add three vectors, the result should be a vector. Thus wouldn't the equation to test linear independence be $a v_1 + b v_2 + c v_3 = (0, 0, 0)$?
Does $0$ represent the integer $0$ or does it represent the zero element, where $a + 0 = a$?
Let $\{v_1,v_2,v_3\}$ be a basis for $\mathbb{R}^3:=\{(x,y,z):x\in\mathbb{R}, y\in\mathbb{R} \text{ and } z \in \mathbb{R}\}$.
As in any vector space E, the "neutral element" or the "origin of E" is denoted $0$ (by analogy with the real number "zero" which has the same property in $\mathbb{R} $); experience shows that there is no disadvantage in denoting by the same sign the neutral element of all vector spaces and the number "zero". So here at $(0,0,0)$, we prefer to simply write $0$.
For example, here, if $a,b$ and $c$ are three real numbers such that $av_1+bv_2+cv_3=0$, then $a=b=c=0$.
But if you understood correctly, in this sentence, the sign $0$ does not designate the same object.