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Mathematical Expression Editor
Challenge Problems for Chapter 6
Argue geometrically to prove that the following transformations are linear:
(a)
Rotation of the plane about the origin through angle .
(b)
Reflection of the plane about the line .
Think in terms of linearity diagrams.
What happens when you rotate two vectors first, then add them, versus adding the
two vectors first, then rotating the sum?
The figure below illustrates the left side of the diagram. Vectors and are added in
the domain, then the sum is rotated through angle .
The next figure illustrates what happens when vectors and are rotated through
angle , then their images are added together.
Because the diagonal of a parallelogram rotates with the parallelogram, it is clear
that
Find the matrix of the linear transformation which rotates every vector in counter
clockwise about the axis when viewed from the positive axis through an angle of 30
and then reflects through the plane.
Click on the arrow to see answer.
Let be a linear transformation such that . Find .
Let be a non-zero vector in . Given any vector in , show there exists a linear
transformation with .
Given in , define by for all in .
(a)
Show that is a linear transformation.
(b)
Show that every linear transformation arises in this way (i.e. for some
in .)
