Change of Basis

Linear Algebra for Machine Learning · Axiom Academy

Converting coordinates between bases Different bases reveal different properties of data. Some bases are more natural or computationally efficient for specific problems. For example, image compression uses bases that capture important visual information efficiently. If we want to convert from basis B₁ to basis B₂, we use a transition matrix P where: The columns of P are the basis vectors of B₂ expressed in basis B₁. Convert coordinates between two 2D bases Original basis B₁: [1, 0], [0, 1] (standard basis) Vector v = [3, 1] in standard basis. What are its coordinates in B₂? We need to solve: [3, 1] = c₁[1, 1] + c₂[1, -1] This gives: c₁ + c₂ = 3 and c₁ - c₂ = 1 So v has coordinates [2, 1] in basis B₂.

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.