Coordinates Relative to a Basis
Express a vector with respect to a chosen basis.
Key idea
Express a vector with respect to a chosen basis.
In depth
Given a basis B, the coordinates of v are the unique weights c with v = c₁b₁ + ... + cₙbₙ. Different bases give different coordinate descriptions of the same vector.
Worked example
B = {(1, 1), (1, −1)}, v = (3, 1): c₁ + c₂ = 3, c₁ − c₂ = 1 → coordinates (2, 1).
Real-world uses
- Describing a position in a rotated or skewed frame, such as a tilted camera view
Common mistake
Mixing up the vector and its coordinates.
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 Coordinates Relative to a Basis?
Express a vector with respect to a chosen basis. It is a topic at the expert level in Linear Algebra.
Where is Coordinates Relative to a Basis used in real life?
Real-world examples: Describing a position in a rotated or skewed frame, such as a tilted camera view.
What should I learn before Coordinates Relative to a Basis?
Make sure you are comfortable with Dimension, the previous topic in Linear Algebra.
What is a common mistake with Coordinates Relative to a Basis?
Mixing up the vector and its coordinates.
How can I practise Coordinates Relative to a 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 Coordinates Relative to a Basis?
Move on to Change of Basis 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.