2x2 Linear Transformations

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All of these files are intended to illustrate visually some basic concepts covered in a Linear Algebra Course. These files were used in the 2008 University of Cincinnati MAT Technology Summer Course.


Image of a Point

This file, Htm.gifThe Image of a Point, illustrates how the coordinates of the preimage (with respect to the original axis) are the same as the coordinates of the image (with respect to the transformed axis). This sketch also illustrates how the the rectangle formed by the preimage is transformed into a parallelogram whose sides lie on the column vectors. The area of this parallelogram is equal to the product of the area of the rectangle and the determinant of the matrix formed by the column vectors. This file also shows the image of the lattice points for reference.

Images of F

This file shows Ggb.gifThe Image of F under different linear transformation. There is a paper worksheet that accompanies this dynamic worksheet Pdf.gifHERE.

Eigenvalues and Eigenspaces

An example of the Htm.gifEigenspaces of 2x2 Matrix transformation.


This dynamic worksheet illustrates the relationship between the Ggb.gifDETERMINANT, the area of the pre-image, and the area of the image.

Solving a System of Equations