Same vector, different coordinates. Learn why choosing the right basis makes everything simpler.
A vector is a geometric object—an arrow in space. But the numbers we use to describe it depend on our choice of basis.
Changing basis is like changing languages: the same object, different descriptions. The right basis can make complicated problems simple.
Every vector can be written as a linear combination of basis vectors.
Different bases → different coefficients (c₁, c₂, ...) → same vector!
Drag the point and watch its coordinates change in both the standard basis (blue/green) and a custom basis (purple/pink). Same location, different numbers!
Drag the orange point. The same point in space has different coordinates depending on which basis you use. The point doesn't move—only how we describe it changes.
To convert coordinates between bases, we use a matrix. P⁻¹ converts from standard to new basis; P converts back.
The vector v in standard coordinates: (2, 1)
In the standard basis, a transformation might look complicated. But in the eigenbasis, the same transformation becomes pure scaling—no shearing!
Key insight: The same transformation looks complicated in the standard basis but becomes pure scaling in the eigenbasis. The off-diagonal terms disappear!
In the eigenbasis, each axis just stretches by its eigenvalue (λ₁ = 3.0, λ₂ = 1.0). This is why eigenvectors are the "natural" coordinates for understanding transformations.
The same ideas extend to 3D. Toggle between standard and custom bases to see how the same point gets different coordinates.
The white point stays in the same location in space, but its coordinates change depending on which basis you use. Drag to rotate the view and see both coordinate systems.
Choosing the right basis is one of the most powerful techniques in linear algebra!