Vector Decomposition: Coordinate Representation via Basis Vectors and Matrix Inv

This GeoGebra example demonstrates how to decompose a planar vector into a linear combination of two basis vectors u and v. By constructing the basis matrix M and computing its inverse Minv via the Invert command, the decomposition coefficients px and py are calculated automatically. The interface features draggable points and vectors, accompanied by grid lines (linesX, linesY) generated with Sequence to visualize the non-orthogonal coordinate grid. Users can dynamically adjust the basis vectors and the target vector, observing in real time how the component line segments, intersection points, and coordinate values change. This interactive exploration deepens understanding of key linear algebra concepts including basis transformation, coordinate representation, and the geometric meaning of matrix operations. Students benefit from immediately seeing the connection between abstract matrix inversion and the concrete decomposition of vectors into components along arbitrary, non-perpendicular axes.