Basis vectors of coordinate systems

I've just begun vector calculus course and I've done coordinate systems so far. But I'm confused about deriving basis vectors in coordinate systems.

I don't understand why we can derive standard basis vectors from the derivatives of the location of point p with respect to x, y and z, i.e. I don't understand e$$_x$$ = $$\partial$$r/$$\partialx$$ ; e$$_y$$ = $$\partial$$r/$$\partialy$$ ; e$$_z$$ = $$\partial$$r/$$\partialz$$ .

Similarly, when we find basis unit vectors of coordinates (u, v, w) I don't understand why the basis unit vectors are

e$$_u$$ $$*h_u$$ = $$\partial$$r/$$\partial u$$

e$$_v$$ $$*h_v$$ = $$\partial$$r/$$\partial v$$

e$$_w$$ $$*h_w$$ = $$\partial$$r/$$\partial w$$

(Here, $$h_u =$$|$$\partial$$r/$$\partial u|$$ , $$h_v =$$|$$\partial$$r/$$\partial v|$$ and $$h_w =$$|$$\partial$$r/$$\partial w|$$ )

I don't understand why do we get basis vectors by partial differentiation. Because, I don't think it would be the case in linear algebra.

(Sorry for terrible formatting though. I'm not familiar with writing maths by computer!)

The fact that they form a basis is reflected in the non-vanishing of the Jacobian at each point $$\mathbf u_0$$ in parameter-space. That is, $$\det D\mathbf r(\mathbf u_0) \neq 0$$ iff $$\frac{\partial \mathbf r}{\partial u}(\mathbf u_0),\frac{\partial \mathbf r}{\partial v}(\mathbf u_0),\frac{\partial \mathbf r}{\partial w}(\mathbf u_0)$$ are linearly independent.
You need to differentiate to see the relationship with linear algebra because $$D\mathbf r(\mathbf u_0)$$ is the natural linear map to associate to the coordinates $$\mathbf r$$ (but it may depend on the point $$\mathbf u_0$$). In particular if $$\mathbf r (x,y,z) = (x,y,z)$$ defines cartesian coordinates, then $$D\mathbf r = I$$ and the columns of $$D\mathbf r$$ are the 'standard basis' of $$\mathbb R^3$$.
More generally, for any invertible linear map $$L$$, the transformation $$\mathbf r(\mathbf x) = L\mathbf x$$ has derivative $$L$$. This map sends the standard basis vectors to the columns $$\mathbf c_i$$ of $$L$$, and note that $$\frac{\partial \mathbf r}{\partial x_i} = \mathbf c_i$$ which matches with change of coordinates from linear algebra.