Fundamentals of Matrix Algebra, 2011a

5.1 Transformaons of the Cartesian Plane
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Figure 5.4: Transforming the Cartesian plane in Example 95
A while ago we asked two quesons. The first was “How do we find the matrix that
performs a given transformaon?” We have just answered that queson (although we
will do more to explore it in the future). The second queson was “How does knowing
how the unit square is transformed really help us understand how the enre plane is
transformed?”
Consider Figure 5.5 where the unit square (with verces marked with shapes as
before) is shown transformed under an unknown matrix. How does this help us understand
how the whole Cartesian plane is transformed? For instance, how can we
use this picture to figure out how the point (2, 3) will be transformed?
y
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x
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Figure 5.5: The unit square under an unknown transformaon.
There are two ways to consider the soluon to this queson. First, we know now
how to compute the transformaon matrix; the new posion of ⃗e 1 is the first column
of A, and the new posion of ⃗e 2 is the second column of A. Therefore, by looking at
the figure, we can deduce that
[ ] 1 −1
A = .
1 2
6
6 At least, A is close to that. The square corner could actually be at the point (1.01, .99).
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