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\)
This interactive diagram provides a means of exploring the effect that a matrix transformation has on the plane as a whole.
Across the top of the diagram are four sliders arranged in a
\(2\times2\) array, which allow a reader to define a
\(2\times2\) matrix
\(A\text{.}\) The matrix transformation
\(T:\real^2\to\real^2\) is given by
\(T(\xvec) = A\xvec\text{.}\)
The bottom part of the diagram consists of two smaller figures, one on the left and one on the right. The figure on the left contains a
\(1\times1\) coordinate grid, a set of labelled axes, the unit square whose lower left vertex is at the origin, and a vector
\(\xvec=\twovec10\text{.}\)
The figure on the right shows the effect of
\(T\) on these features. In particular, the vector
\(T(\xvec)\) is shown along with the transformed coordinate grid and unit square.
Instructions.
The diagram below demonstrates the effect of a matrix transformation
\(T\) on the plane. You may modify the matrix
\(A=\begin{bmatrix} a \amp b \\ c \amp c
\\ \end{bmatrix}\) defining
\(T\) through the sliders at the top.
Since a matrix transformation takes a vector as input and produces a vector as output, we will show the inputs and outputs on separate sets of axes. In particular, the axes on the left represent the inputs while the axes on the right illustrate how input features are transformed by
\(T\text{.}\)
