Same vector, different coordinates. Learn why choosing the right basis makes everything simpler.
A vector is an arrow, but the numbers that describe it depend on your choice of basis. Same object, different descriptions, and the right basis can make a hard problem easy.
Drag the orange point and read its coordinates in 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.
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!
Press Next and Prev to step a vector between coordinate systems. P⁻¹ takes standard coordinates to the new basis; P takes them back.
The vector v in standard coordinates: (2, 1)
To convert coordinates between bases, we use a matrix. P⁻¹ converts from standard to new basis; P converts back.
P matrix: Columns are the new basis vectors expressed in standard coordinates
Switch between Standard Basis and Eigenbasis. In the eigenbasis the same transformation is pure scaling, with no shear.
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.
In the standard basis, a transformation might look complicated. But in the eigenbasis, the same transformation becomes pure scaling—no shearing!
Choosing the right basis is one of the most powerful techniques in linear algebra!
Toggle between Standard Basis and Custom Basis, and drag to rotate the view. The same point gets different coordinates in each.
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.
Spot a mistake or have an idea?