| Question:If I have the matrix for a reflection-dilation, how can I
find ascertain the angle of the line we are using as the reflection axis?
|
Answer:
A reflection-dilation matrix is of the form
A = | a b |
| b -a |
The dilation factor is the length of the first column vector
which is (a^2+b^)^1/2.
Dividing the matrix by the dilation factor gives
B = | cos(2alpha) sin(2 alpha) |
| sin(2alpha) -cos(2alpha) |
Special cases are
a=1,b=0 Reflection at the x axes
a=-1,b=0 Reflection at the y axes
a=0, b=1 Reflection at x=y
To get the line is to look at the first
column vector of the matrix then half the angle between
this vector and the x-axes.
|
| Question:What exactly is the difference between the equations of a damped
harmonic oscillator and a driven harmonic oscillator?
|
Answer:
A damped harmonic oscillator is the linear system
f''(t) + f'(t) + f(t) = 0
a driven harmonic oscllator is an oscillator with an
inhomogeneous driving like
f''(t) + f(t) = sin(3t)
You can also drive a dampted oscillator
f''(t) + f'(t) + f(t) = sin(3t)
|
| Question:What are "Kyle numbers"?
|
Answer:
The book (and especially the solution book) mentions Kyle numbers.
They are a short cut to compute the kernel, especially for 2 times
2 matrices. If a 2 x 2 matrix is not invertible, the two columns have
to be parallel. For
A = | 2 -4 |
| 4 -8 |
for example, we see that the second column is -2 times the first column.
The Kyle numbers are 1, 2 because adding the first to 2 times the second
column gives zero. So that [1,-2]^T is in the kernel.
Its better to stick to row reduction to compute the kernel since it is
easy (especially for larger matrices) to get only part of the kernel.
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