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Final

MATH 33AH Lecture Notes - Lecture 3: Linear Map, Glossary Of Video Game Terms, Formula AtlanticExam


Department
Mathematics
Course Code
MATH 33AH
Professor
All
Study Guide
Final

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Math 33A Discussion Example
Austin Christian
October 23, 2016
Example 1. Consider tiling the plane by equilateral triangles, as below.
Let vand wbe the orange and green vectors in this figure, respectively, and let B=
{v,w}be the basis for R2formed by these two vectors. Let ube the purple vector.
(a) Sketch the vector 1
2B
in the figure.
(b) Give the components of uwith respect to the basis B.
(c) Let T:R2R2be the linear transformation that rotates the plane counterclockwise
through an angle of 2π/3. Find the matrix B[T]B.
(d) Is the terminal point of the vector 1008
2016 B
in the center of one of the red hexagons
or on an edge?
(Solution)
(a) Because this vector is given in the basis B={v,w}, the desired vector is equal to
1v+ 2w, which is the blue vector in the following figure:
1

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(b) We see that we can get to the terminal point of uby following wand then following
vtwice. Alternatively, we could follow vtwice and then follow w. In either case,
we see that
u=w2v=2
1B
.
(c) When we apply this rotation, vand ware carried to their dashed counterparts in the
following figure:
We see that T(v) = wvand T(w) = v, so
T1
0B=1
1B
and T0
1B=1
0B
.
This means that the matrix of Twith respect to Bis given by
B[T]B=11
1 0 .
(d) Notice that the vector 1
2B
lies in the center of a hexagon:
It is not difficult to see that integer multiples of this vector will also lie in the centers
of various hexagons. Because our vector is such a multiple, it lies in the center of the
1008-th hexagon above the hexagon containing the origin.
2
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