Transcript 2.1 Linear Transformations and their inverses day 2

2.1 Day 2 Linear Transformations
and their Inverses
For an animation of this topic visit:
http://www.ies.co.jp/math/java/misc/don_trans/don_trans.html
A=
Multiply the coordinates
of the light blue dog by
A to get the dark blue dog
Note: view each coordinate separately as the equation Ax=b
If the coordinates (of the dog) are put together to make a matrix,
the equation takes Form AB= C
Example
To encode the position of the dog on the
previous slide multiply each coordinate of
the dog by the matrix in the form Ax = b
(A is the matrix)
x is a vertical vector with components of the
coordinates of the dog.
b is the new coordinate.
This is called a transformation matrix
Example
If we want to start with the dark blue dog and
end with the light blue dog we will need to
multiply the coordinates of the dark blue
dog by a matrix. This matrix is called the
inverse of Matrix A.
We learned how to compute an inverse
yesterday.
Find the inverse
What if we had multiplied the coordinates of
the dog by the following matrix
What would the inverse be?
What would the transformation look like?
(answer on next slide)
The graph
What would the inverse of
this matrix be? Why?
Problem 13 a
Prove that for the given matrix, it is invertible if and
only if the determinant is not = 0
Hint prove the case of a = 0 and
separately
Problem 13 solution
Problem 13 solution continued
Linear Transformations
This represents a transformation from R3 to R
Linear transformations
If there is a n nxm matrix A such that
T(x) = A(x)
For all x in
Problem 2
Is this transformation linear?
What is the matrix of transformation?
How could this transformation be changed to
make it non-linear?
Problem 2 Solution
Possible solution to make system non-linear
y1 = x1y2 (is non-linear due to multiplication of variables)
y2 = x2 +1 (is non-linear due to adding a constant
(note: this is different from other courses you have taken)
y3= (x1)2 (non-linear due to an exponent)
If any one of these is non-linear than the entire expression is
non-linear
Problem 16
Describe the transformation.
And the inverse of the transformation
Note: when you do your homework keep a
record of each type of transformation
Problem 16 Solution
Rectangular matrices
A 3 x 3 matrix represents a transformation
from R3 to R3
What kind of transformation does a 2 x 3
matrix represent?
What kind of transformation does an mxn
matrix represent?
Rectangular matrices
A 3 x 3 matrix represents a transformation
from R3 to R3
What kind of transformation does a 2 x 3
matrix represent? R3 to R2
What kind of transformation does an mxn
matrix represent? Rn to Rm
Homework
p. 50 1-15 all (13 was already done as part of the lesson)
16-23 all (May use the website shown on first slide to help
with the geometric interpretation)
Some math quotes:
The Romans didn't find algebra very challenging, because X
was always 10.
(Aaron Dragushan)
“There are three kinds of lies: lies, damned lies, and statistics.”
Attributed by Mark Twain to Benjamin Disraeli.