Coordinate Systems

Linear Algebra (Matrices) · Axiom Academy

LESSON Coordinates Relative to a Basis A basis is the grid you measure against — switch the grid and the same vector gets a whole new coordinate readout. 1. A Coordinate Vector Is a List of Weights Fix a basis . Because the basis vectors are independent and span the plane, every vector can be built from them in exactly one way — there is a single pair of weights c_1,c_2 with . The coordinate vector is just that list of weights. a unique linear combination of the basis vectors the coordinate vector lists those weights Here , . The animation lays down then to reach , so the weights are 1 and 1 : 2. Same Vector, Different Coordinates The standard basis is only one choice of ruler. Keep the arrow fixed in place and swap the grid underneath it: read it against the square grid and you get (3,3) ; read the identical arrow against a skewed grid and you get (2,1) . The vector didn't move — the grid you measured it with changed. Both readouts describe the same . Plugging the skewed weights back in confirms it lands on the same point: You've seen that coordinates are always measured against a basis — change the basis and the same vector reads differently, even though the vector itself never moves. Scroll up to revisit any step.

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