The determinant is the test
You do not have to eyeball it. The readout gives the determinant, which is the signed area of that parallelogram, and it reaches zero exactly when the span collapses.
That is the same number the [determinant module](determinant.html) covers as an area scale factor, seen from another side. A matrix whose determinant is zero squashes the plane onto a line, which is precisely why it has no inverse: several inputs land on the same output, so nothing can undo it.
Basis
A basis is a set of vectors that spans the space and is linearly independent — enough to reach everything, with nothing redundant.
Both halves matter. Too few vectors and you cannot reach everything. Too many and the representation stops being unique, because there is more than one way to write the same point. A basis is the exact amount.
Once you have one, every point in the space has exactly one set of coordinates in it. That is the whole reason coordinates work at all, and it is why "[change of basis](matrix_as_transformation.html)" is a meaningful operation rather than a rearrangement.
The plane needs exactly two vectors in any basis; three-dimensional space needs three. That count is the dimension, and it does not depend on which basis you choose.
Orthogonality
Two vectors are orthogonal when their [dot product](vectors_and_dot_product.html) is zero — the algebraic way of saying they meet at a right angle. Drag until the readout says so.
Orthogonality is not required for a basis. The default pair above is a perfectly good basis and is nowhere near perpendicular. But an orthogonal basis is worth reaching for, because finding the coordinates of a point becomes a matter of taking one dot product per axis instead of solving a system of equations.
An orthonormal basis adds that every vector has length 1. Then the matrix whose columns are those vectors has a remarkable property: its inverse is its transpose. Undoing the transformation costs nothing at all, which is why orthonormal bases turn up wherever numerical stability matters.
The standard basis — (1, 0) and (0, 1) — is orthonormal, which is why ordinary coordinates feel so natural.
Where this shows up
PCA finds an orthonormal basis aligned with the directions of greatest variance in the data. The orthogonality is the point: the components do not overlap, so each one describes something the others do not.
[SVD](singular_value_decomposition.html) produces two orthonormal bases, one for the input space and one for the output.
QR decomposition exists to manufacture an orthonormal basis from an arbitrary one, and least-squares solvers use it because the resulting system is far better conditioned than the direct approach.
Fourier and wavelet transforms are changes into an orthogonal basis chosen so the interesting structure shows up in a few coordinates.
Where it goes wrong
Nearly dependent vectors. A determinant of exactly zero is easy to spot; a determinant of 0.001 is the same problem wearing a disguise. The basis still technically works, and everything computed in it is numerically unstable. The [condition number](singular_value_decomposition.html) is how you measure it.
Assuming orthogonal means independent. It does — orthogonal non-zero vectors are always independent. The converse fails, and that is the direction people assume.
Forgetting to normalise. Orthogonal gives you the easy inverse only when the vectors also have length 1.