Change of Basis Matrix Calculator
Convert coordinates from Standard Basis to a Custom Basis B.
1. Define New Basis B
2. Vector x (Standard Coords)
Change of Basis Matrix (P)
Coordinates in Basis B: [x]B
Basis Grid Visualization
Visual: The black dot is your vector. The skewed green lines show the new “grid”. The path shows how to get to the dot using v1 and v2.
What is a Change of Basis?
Imagine a “basis” as a type of graph paper. The standard basis is your regular square grid paper. But what if you had graph paper where the lines were slanted or spaced differently? That’s a new basis.
A vector is just a point in space. However, the coordinates we use to describe that point depend on which graph paper we are using.
The “Language” Analogy
Think of the vector as an object (e.g., a Cat).
In Standard Basis (English): We call it “Cat”.
In Basis B (French): We call it “Chat”.
It’s the same animal, just a different label. The Change of Basis Matrix is the dictionary that translates between them.
The Formula
Let P be the matrix formed by the new basis vectors.
To go from New coordinates to Standard: x = P · [x]B
To go from Standard to New coordinates: [x]B = P-1 · x
Why do we do this?
In physics and computer graphics, choosing the right basis can make a hard problem easy. For example, if you are analyzing a spinning object, it’s much easier to use a coordinate system that spins with the object than to use a stationary one.
Frequently Asked Questions (FAQ)
Q: What if the Determinant is 0?
A: If det(P) = 0, your basis vectors are “linearly dependent” (they lie on the same line). They cannot form a valid basis for 2D space because they flatten the grid into a line. You cannot convert coordinates uniquely.
Q: Is the Change of Basis Matrix always square?
A: Yes, if we are changing bases within the same dimension (e.g., 2D to 2D). We need exactly 2 basis vectors to define a 2D space.
Q: How does this relate to Diagonalization?
A: Diagonalization is a specific type of basis change where we switch to the “Eigenbasis”. In that basis, matrix operations become simple multiplications (stretching) without rotation.