Change of Basis
Convert coordinates between bases.
Key idea
Convert coordinates between bases.
In depth
The change-of-basis matrix P (columns are the new basis vectors) satisfies v = P[v]_B. To go the other way use P⁻¹. A transformation matrix in the new basis is P⁻¹AP.
Worked example
With P = [1 1; 1 −1], coordinates (2, 1) correspond to the vector P(2, 1) = (3, 1).
Formula
Real-world uses
- Converting between measurement frames in robotics, physics and image compression
Common mistake
Using P instead of P⁻¹, or vice versa.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Change of Basis?
Convert coordinates between bases. It is a topic at the expert level in Linear Algebra.
What is the formula for Change of Basis?
The key formula is [v]_C = P [v]_B.
Where is Change of Basis used in real life?
Real-world examples: Converting between measurement frames in robotics, physics and image compression.
What should I learn before Change of Basis?
Make sure you are comfortable with Coordinates Relative to a Basis, the previous topic in Linear Algebra.
What is a common mistake with Change of Basis?
Using P instead of P⁻¹, or vice versa.
How can I practise Change of Basis?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Change of Basis?
Move on to Column Space and Row Space next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.