Linear Algebra
Change of Basis
The same arrow, described in two different grids, gets two different pairs of numbers. The matrix of basis vectors is the dictionary between them, and the pairs it can produce are exactly the span of the basis.
b₁b₂va·b₁ + b·b₂the figure FA F, applied directly
Drag the tips of b₁, b₂ and v
v in their coordinates, [v]B—
—
View
Ruler
A′, the simple map in their grid
FP⁻¹ · A′ · P
Their first basis vector b1
Their second basis vector b2
The vector v, in our coordinates
Presets for the basis
Values
det P—
P⁻¹—
A′, their grid—
A = P A′ P⁻¹, our grid—
What to observe
- Switch to their grid: the grid bends onto b₁, b₂ but v never moves. The walk 5 along î, 1 along ĵ becomes 2 along b₁, −1 along b₂, andP = [b₁ b₂] turns (2, −1) back into (5, 1).
- On Stretched, b₁ is twice as long as î, so v needs 2.5 of them instead of 5, while the half-length b₂ turns 1 into 2. A longer ruler gives a smaller number.
- In Span view, line b₁ and b₂ up: every walk stays on one line, det P = 0, and v, off that line, has no coordinates at all.
- Animate A = P A′ P⁻¹: translate into their grid (P⁻¹), stretch along b₁ (A′ is diagonal there), translate back (P). The F lands exactly on A F, four messy numbers in our grid.
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