Interactive Linear Algebra
Dan Margalit & Joseph Rabinoff · Georgia Tech · Open textbook
Read §2.1 on vectors and §2.8 on bases as coordinate systems.
Open the sourceCoordinates are not the arrow
Vectors as movements; coordinates as a chosen language.
Prerequisites: 02 · Why enlarge the number system?, 04 · Functions and what they forget
Understand
Objectives
- Express one fixed vector in two explicitly named bases.
- Recover a vector by combining its new basis arrows.
- Reject a pair that fails to span the plane.
Start with something you can see
Draw a displacement two units right from the origin. Keep the arrow fixed, then replace the horizontal and vertical measuring arrows by b₁=(1,1) and b₂=(1,−1). Predict its new coordinate pair: one of each new arrow adds to (2,0). The MIT change-of-basis excerpt begins at 38:00. [1]
Give the idea a precise name
A vector is the displacement, while a coordinate pair records its coefficients in a named basis. In ℝ², addition and scaling are componentwise. The standard basis is e₁=(1,0), e₂=(0,1). The pair b₁=(1,1), b₂=(1,−1) is another basis because its members are independent and span ℝ². For any v in this plane, its b-coordinates (c₁,c₂) are the unique coefficients in the equation:
v = c₁b₁ + c₂b₂
The letters b₁,b₂ denote the chosen arrows, and c₁,c₂ are scalars. Change the basis, and the coefficients generally change even when v remains fixed. Write the basis alongside a coordinate list; without it, (1,1) can name different vectors.
Work one small world
For v=(4,2) in the standard basis, solve c₁+c₂=4 and c₁−c₂=2. Thus c₁=3 and c₂=1, and 3b₁+b₂=(4,2). The same arrow has standard coordinates (4,2) and b-coordinates (3,1). For (2,0), its b-coordinates are (1,1).
| Fixed arrow | Standard coordinates | b-basis coefficients |
|---|---|---|
| v | (2,0) | (1,1): b₁+b₂ |
| w | (4,2) | (3,1): 3b₁+b₂ |
Counterexample to “any two arrows make a basis”: (1,0) and (2,0) are distinct but parallel. No combination of them gives (0,1), so there is no two-coordinate description of that arrow using this pair. A proposed change of basis must actually be invertible.
THE BRIDGEA matrix describes an action only after its input and output coordinate systems are named.
WHERE THIS IDEA STOPSPoints and displacements need an origin to share coordinate notation. A coordinate list alone is not an intrinsic vector.
Experiment
Keep v=(2,0) fixed. First predict its b-coordinates, then add b₁ and b₂ on paper and compare the result with the diagram and table.
Can an arrow have coordinate (1,1) in one basis and (2,0) in another?
Check & explain
With b₁=(1,1) and b₂=(1,−1), what is the first b-coordinate of v=(6,−2)?
Which remains unchanged under an invertible change of basis?
- The coordinate pair of every vector
- The vector and its dimension
- The chosen basis arrows
Teach back
Describe v=(4,2) in the standard and b-bases, verify both combinations, and explain why two parallel arrows cannot replace b₁,b₂.
- Name each basis.
- Verify 3b₁+b₂=(4,2).
- Use failure to span for the parallel pair.
Read deeper
Mathematics for Machine Learning
Deisenroth, Faisal & Ong · Freely readable textbook
Read §2.4 on vector spaces, printed p.35 onward, as a second pass.
Open the source31. Change of Basis; Image Compression
Gilbert Strang · MIT OpenCourseWare · Lecture · 50 min
Start at 35:00. Watch the short core 38:00–42:00; then the longer passage 35:00–50:00. The full 50-minute lecture is available from the same link.
Open the source