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GEOMETRIC EFFECT
Theorem:
Given T:R2 R2 is multiplication by an invertible matrix
Then the geometric effect of T is nothing but the
appropriate succession of shears , compressions ,
expansions and reflections.
EG 1:
A = K1 0
0 K2
Here A has a
geometric
expansion or
compression of k1
in the x- direction
and k2 in the y-
direction
EG 2:
1 2
3 4
Express A = As a product of elementary matrices and then describe the geometric effect of
multiplication by A in terms of shears , compressions , expansions and reflections.
THEOREM 2:
a) The image of a straight line is a straight line.
b) The image of a straight line through the origin is a straight line through the origin.
c) The images of parallel straight lines are parallel straight lines.
d) the image of line segment joining point P and Q is the line segment joining the images of
P and Q.
e) The images of three points lie on a line if and only if the points themselves lie on a line.
Express
A =
as a product of elementary matrices , and then describe the geometric
effect of multiplication by A in terms of shears , compressions , expansions ,
and reflections.
1 2
3 4
The image is a parallelogram.
According to Theorem , the invertible matrix
maps the line y = 2x +1 into another line. Find its equation
Y = 2X +1
3
1
2 1
Geometric effect

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Geometric effect

  • 2. Theorem: Given T:R2 R2 is multiplication by an invertible matrix Then the geometric effect of T is nothing but the appropriate succession of shears , compressions , expansions and reflections.
  • 3. EG 1: A = K1 0 0 K2 Here A has a geometric expansion or compression of k1 in the x- direction and k2 in the y- direction
  • 4. EG 2: 1 2 3 4 Express A = As a product of elementary matrices and then describe the geometric effect of multiplication by A in terms of shears , compressions , expansions and reflections.
  • 5. THEOREM 2: a) The image of a straight line is a straight line. b) The image of a straight line through the origin is a straight line through the origin. c) The images of parallel straight lines are parallel straight lines. d) the image of line segment joining point P and Q is the line segment joining the images of P and Q. e) The images of three points lie on a line if and only if the points themselves lie on a line.
  • 6. Express A = as a product of elementary matrices , and then describe the geometric effect of multiplication by A in terms of shears , compressions , expansions , and reflections. 1 2 3 4
  • 7. The image is a parallelogram.
  • 8. According to Theorem , the invertible matrix maps the line y = 2x +1 into another line. Find its equation Y = 2X +1 3 1 2 1