2.2 Linear Transformation in Geometry Example. 1 Consider a linear ...
2.2 Linear Transformation in Geometry Example. 1 Consider a linear ... 2.2 Linear Transformation in Geometry Example. 1 Consider a linear ...
Example. 5 Give a geometric interpretation of the linear transformation. T (x) = a −b b a Rotation-dilations A matrix with this form a −b b a denotes a counterclockwise rotation through the angle α followed by a dilation by the factor r where tan(α) = b a and r = a2 + b2 . Geometrically, ✻ ✡ α ✡ ✡ ✡ ✡✡✡✡✣ ✲ a b = x r cos α r sin α ✲ 4
[Shears] Example. 6 Consider the linear transformation T (x) = 1 1 2 0 1 To understand this transformation, sketch the image of the unit square. x 1 1 2 Solution The transformation T (x) = 0 1 is called a shear parallel to the x1-axis. 5 x
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<strong>Example</strong>. 5 Give a geometric <strong>in</strong>terpretation<br />
of the l<strong>in</strong>ear transformation.<br />
T (x) =<br />
<br />
a −b<br />
b a<br />
Rotation-dilations A matrix with this form<br />
<br />
a −b<br />
b a<br />
denotes a counterclockwise rotation through<br />
the angle α followed by a dilation by the factor<br />
r where tan(α) = b <br />
a and r = a2 + b2 . Geometrically,<br />
✻<br />
✡ α<br />
✡<br />
✡<br />
✡ ✡✡✡✡✣<br />
✲<br />
a<br />
b<br />
<br />
<br />
=<br />
<br />
x<br />
r cos α<br />
r s<strong>in</strong> α<br />
✲<br />
<br />
4
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