I'm currently reading Spivak's Calculus on Manifolds and I'm confused about how practically figure out orientations.
I understand that in $\mathbb{R^3}$, given two different bases $e_1, e_2, e_3$ and $f_1, f_2, f_3$, we can just check if the $3 \times 3$ matrix $E$ and $F$ which consist of $e_1, e_2, e_3$ and $f_1, f_2, f_3$ as columns have the same determinant.
However, I'm not sure how to do this for manifolds, since the dimension of the $k$-manifold could be less than $n$.
Question 1: Suppose I had a 2-manifold $M$ in $\mathbb{R^3}$, and a tangent points $x$ in $M$. If a basis for the tangent space at $x$ is (2,0,4), (0,0,5) and another is (-1,0,3), (1,0,0), what are the steps computationally to check that the orientations are consistent?
Question 2: Now suppose I have two different points on the (connected) manifold $x$, $y$ in $M$. If a basis for the tangent space at $x$ is (2,0,4), (0,0,5) and a basis for the tangent space at $y$ is (2,2,3), (1,0,0), what are the steps computationally to check that the orientations are consistent? Are the coordinate patches whos image contains both points required?
(Note: Although a vector space "basis" often refers to a set of vectors without regard to order, Spivak and this question and answer are here using "basis" as a sequence of vectors, where order is significant.)
For question 1, Spivak doesn't actually talk about "consistent orientations" for different bases of the same tangent plane. The first discussion of orientation (Chapter 4, p. 82) just defines an orientation for a single given vector space, as either of the two sets of bases with positive-determinant matrixes converting between bases in the same set, and negative-determinant matrixes converting to a basis in the other set. When applied to manifolds (Chapter 5, pp. 117-119), Spivak talks about choosing a single orientation (set of bases) for each tangent plane $M_x$, and then defines "consistent orientation" as a potential property of that collection of orientations. But yes, to stick with Spivak's terminology, at a single point $x$ we can ask if $[(2,0,4),(0,0,5)] = [(-1,0,3),(1,0,0)]$. That is, are the two bases elements of the same orientation?
As already described in comments, one simple way would be to write one basis in terms of the other, then test the sign of the transformation determinant.
$$ \begin{align*} (2,0,4) &= \frac{4}{3}(-1,0,3) + \frac{10}{3}(1,0,0) \\ (0,0,5) &= \frac{5}{3}(-1,0,3) + \frac{5}{3}(1,0,0) \\ \det T &= \left| \begin{array}{cc} \frac{4}{3} & \frac{10}{3} \\ \frac{5}{3} & \frac{5}{3} \end{array} \right| < 0 \end{align*} $$
No, these bases are not in the same orientation.
Or to work in $\mathbb{R}^2$ instead of the subspace of $\mathbb{R}^3$, we could use any "coordinate system" (sometimes in other sources called "chart") at $x$, a differentiable injective function $f$ from an open set in $\mathbb{R}^2$ to $M \subset \mathbb{R}^3$. Then we can look at the 2-dimensional vectors $f_*(v)$. In this example, since $M_x$ is essentially the $(x_1, 0, x_3)$ plane, a decent coordinate system might be $f(a,b) = (x_1 + a, \kappa(a,b), x_3 + b)$ where $\kappa$ describes the $x_2$-coordinate of the manifold in a neighborhood of $x$. Then $f_* \big((2,0,4)\big) = (2,4)$ etc. and the calculations turn out essentially the same. Maybe in some examples with a slanted tangent space, this method could end up simpler.
For question 2, yes, when we consider multiple points of the manifold, we have to use the coordinate systems / charts to relate almost any properties of the points or their tangent spaces, including orientation. A consistent choice of orientations requires that for all pairs of points $x$ and $y$ in the domain of a coordinate system $f$, the pullback of the usual orientation of $\mathbb{R}^3$ matches both or neither of the orientations chosen for $M_{f(x)}$ and $M_{f(y)}$:
$$ [f_*((e_1)_x), f_*((e_2)_x)] = \mu_{f(x)} $$ if and only if $$ [f_*((e_1)_y), f_*((e_2)_y)] = \mu_{f(y)} $$
So question 2 can't be answered without knowing more about the manifold. Different manifolds including points $x$ and $y$ might "twist" in different ways, have the same tangent spaces at $x$ and $y$, but have different results about orientations of those tangent spaces.
Often when we describe a concrete manifold embedded in $\mathbb{R}^n$, we do it with parameterized equations - which essentially means we start out knowing at least one coordinate system / chart onto most of the manifold. If these equations are differentiable on the chart's domain, the pullback of the usual orientation of $\mathbb{R}^n$ then automatically gives us an obvious choice of orientation which is consistent on the image of that chart. So then to see if the manifold is orientable, we just need to check the "glued" or "limit" spots not covered by that primary chart, to see if the nearby chosen orientations will allow consistent choices for the remaining tangent spaces.