Linear transformations onto vs one to one
The transformation preserves magnitutes and maps vectors along the axis to themselves. One such linear transformation is represented by the matrix, which maps vectors along the axis to vectors along the axis (that is, maps the greeen points on the axis to black points on the axis). The symbol stands for 3×3 matrices representing linear transformations, and their matrix entries are obtained by assigning values to the symbols. The black points are the images of linear transformations of the green points. The red vector, v, corresponds to the linear combination of the vectors, , and. The black vector, u, corresponds to the linear combination of the vectors and. The green points are the linear combinations of the vectors, , and for scalars, , and obtained by setting the sliders, and and using values for scalars ranging from to. The brown dot represents a point whose component values are assigned using the symbols, , and. The light blue line represents the vector whose component values are assigned using the symbols, , and. The purple line represents the vector whose component values are assigned using the symbols, , and. The blue line represents the vector whose component values are assigned using the symbols, , and. For further explanations and the definition of the controls, see below the descriptions of the three Snapshots. This Demonstration can be used to support linear algebra activities on vector space concepts: geometric vector, linear combination, linear independence, spanning set, span, basis, coordinates, and so on and topics on linear transformations: domain, range, kernel (null space), orthogonal projection, one-to-one and onto transformations.