Linear Algebra · Advanced · 7 min
Basis & coordinates
Coordinates describe a vector relative to a chosen independent set of directions.
WHAT HAPPENS IF…For t=−1, what are the coordinates of (4,2)?
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For v=(4,2), b₁=(1,1), b₂=(1,t), the equations are α+β=4 and α+tβ=2. When t≠1, β=−2/(t−1) and α=4−β. As the directions nearly coincide, the coordinates can become large even though the vector remains small.
A detail that matters
Dotting with basis vectors gives coordinates only for an orthonormal basis, not an arbitrary basis.
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