Basis and coordinates
Objective. Build bases and express vectors in coordinates.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A basis of a vector space V is a linearly independent set of vectors that also spans V, so every vector in V can be written uniquely as a linear combination of basis vectors, called its coordinates relative to that basis. Building a basis typically starts from a spanning set and removes dependent vectors, or starts from an independent set and adds vectors until it spans. For ℝ², the standard basis {(1,0),(0,1)} gives coordinates equal to the vector's own entries, but for basis {(1,1),(1,−1)}, a vector like (4,2) is written as c₁(1,1)+c₂(1,−1)=(4,2), solved to give c₁=3, c₂=1, so its coordinates in this basis are (3,1). Coordinates always depend on the chosen basis, not just the vector itself.
Worked example
Find the coordinates of the vector (4,2) relative to the basis {(1,1),(1,−1)}.
- 1Write (4,2)=c₁(1,1)+c₂(1,−1), giving c₁+c₂=4 and c₁−c₂=2.
- 2Add the two equations: 2c₁=6, so c₁=3.
- 3Substitute into c₁+c₂=4: 3+c₂=4, so c₂=1.
- 4Check: 3(1,1)+1(1,−1)=(3+1,3−1)=(4,2), confirming coordinates (3,1).
Answer: (3, 1)
Concept mastery
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Practice
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Start practisingCommon mistake. Students assume a vector's coordinates are the same in every basis, writing (4,2) as its own coordinate pair even when a non-standard basis is specified, instead of solving the linear combination equation for that particular basis.
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