This java applet is a simulation that demonstrates some properties of
matrices
and how they can be used to describe a linear transformation in
two dimensions.
The applet shows two vectors, one red and one blue. The red vector
is the first column of a matrix (shown on the right side of the
screen). The blue vector is the second column.
Since the red vector is the first column of the matrix, it is equal to
the image of the (1,0) vector under the transformation described
by the matrix. The blue vector is the image of the (0,1) vector.
When the applet starts up, the matrix is simply the identity matrix,
so the red vector is (1,0) and the blue vector is (0,1).
The applet also shows a big green Z with a line under it and a green
box around it. This is just a simple design that shows the action
of the matrix. The applet transforms the design using the matrix
before drawing it. As the matrix changes, the image of the design
changes. The Z doesn't have any significance other than the fact that
it is simple to draw and it is asymmetrical, so you can tell if it's
backwards.
To change the matrix, move one of the two vectors by clicking on the
arrowhead and dragging it. Alternatively you can click and drag anywhere
on the grid to rotate the two vectors or modify their lengths. This will
move the vectors around, which will change the matrix. This causes
the design to change. Since the matrix is a linear transformation,
all the lines in the design will remain straight, and the Z will always
be centered at the origin.
As you move the mouse around the grid you may see a gray arrow pointing
from the mouse position to a purple dot. (Or you may just see a
purple dot.) This shows the image of the point under the mouse,
when multiplied by the matrix. For example if you move the mouse to
(1,0) (which is where the red arrow originally points) you will
see a purple dot under the red arrowhead. That means that the image
of (1,0) is the same as where the red arrow is pointing. Also if you
move the mouse around where the green square originally was, or where
the green Z was, you will
see the purple dot move around the present location of the green square or Z.
If you want to change the elements of the matrix to a certain set of values,
you can
just drag the red and blue vectors, since the red vector is the first
column and the blue vector is the second column. For
example if you want to set
the matrix to:
0 1
1 2
then you just drag the red vector to (0,1), and drag the blue vector
to (1,2).
To reset the matrix to the identity matrix, you can click the "Identity"
button. To replace the matrix with its transpose (which means that
you interchange the upper right and lower left elements), click
"Transpose". To invert the matrix, click "Invert"; this only works
if the matrix is invertible, which is only true when the determinant
is nonzero.
To rotate the vectors 30 degrees clockwise or counter-clockwise
(which involves multiplying
the matrix by a 30-degree rotation matrix) click "Rotate CW" or
"Rotate CCW".
To reflect the vectors across the X axis click "Reflect X".
To reflect the vectors across the Y axis click "Reflect Y".
Below the matrix you will see some additional information displayed about it.
"det M" is the determinant of the matrix. "tr M" is the trace.
And lambda1 and lambda2 are the two eigenvalues. Notice that the
sum of the two eigenvalues is the trace, and the product of them is
the determinant.
You may see an orange and/or a yellow vector drawn on the grid. These
are the two eigenvectors. The yellow one is associated with lambda1 and
the orange one is associated with lambda2. When you move the mouse to
where an eigenvector is pointing (or any point along the same line as
the eigenvector), you will see that the purple dot (the image of that point)
is co-linear with the eigenvector. That means that the image of that
point is simply that point multiplied by the eigenvalue. If the
eigenvalue is zero or very small then the image will be at or near the
origin. If the eigenvalue is negative then the image will be on the
other side of the grid, through the origin. If the eigenvalue is
positive but less than one, then the image will be closer to the origin
but still along the same line; if it is greater than one then it will
be farther away.
If you do not see one or both of the yellow and orange vectors,
there may be one of several
reasons. If the matrix is set to the identity matrix (or a scalar multiple),
then no eigenvectors are shown because there are no two unique
eigenvectors; any vector is an eigenvector.
If the eigenvalues are complex, then the eigenvectors are complex too,
so we can't show them on a simple 2-d grid. If the two eigenvalues
are the same, then there may only be one eigenvector in some cases.
For example, this matrix has only one eigenvector:
2 0
2 2
In other cases, there should be two eigenvectors, but you may not be able
to see one or both of them because the red or blue vectors may be drawn on
top of them. The eigenvectors always have a length of 1, so if the
red and/or blue vectors also have a length of 1, then you may not be able to
see or or both of them. To fix this, change the length of the red
and blue vectors slightly.
You may notice dark blue and dark red dots moving around on the grid. The
dark red dot is the image of (1,0) under the inverse of the
matrix. And the dark blue dot is the image of (0,1) under the
inverse. So if you move the mouse to the dark red dot, you will see
a purple dot at (1,0). And if you click the "Invert" button, you will
see the red vector move to where the dark red dot is, and vice versa.
The source. java@falstad.com
A far higher place must be assigned to Judaism among the competitors for the allegiance of Europe. The cosmopolitan importance at one time assumed by this religion has been considerably obscured, owing to the subsequent devolution of its part to Christianity. It is, however, by no means impossible that, but for the diversion created by the Gospel, and the disastrous consequences of their revolt against Rome, the Jews might have won the world to a purified form of their own monotheism. A few significant circumstances are recorded showing how much influence they had acquired, even in Rome, before the first preaching of Christianity. The first of these is to be found in Cicero’s defence of Flaccus. The latter was accused of appropriating part of the annual contributions sent to the temple at Jerusalem; and, in dealing with this charge, Cicero speaks of the Jews, who were naturally prejudiced against his client, as a powerful faction the hostility of which he is anxious not to provoke.330 Some twenty years later, a great advance has been made. Not only must the material interests of the Jews be respected, but a certain conformity to their religious prescriptions is considered a mark of good breeding, In one of his most amusing satires, Horace tells us how, being anxious to shake off a bore, he appeals for help to his friend Aristius Fuscus, and reminds him of217 some private business which they had to discuss together. Fuscus sees his object, and being mischievously determined to defeat it, answers: ‘Yes, I remember perfectly, but we must wait for some better opportunity; this is the thirtieth Sabbath, do you wish to insult the circumcised Jews?’ ‘I have no scruples on that point,‘ replies the impatient poet. ‘But I have,’ rejoins Fuscus,—‘a little weak-minded, one of the many, you know—excuse me, another time.‘331 Nor were the Jews content with the countenance thus freely accorded them. The same poet elsewhere intimates that whenever they found themselves in a majority, they took advantage of their superior strength to make proselytes by force.’332 And they pursued the good work to such purpose that a couple of generations later we find Seneca bitterly complaining that the vanquished had given laws to the victors, and that the customs of this abominable race were established over the whole earth.333 Evidence to the same effect is given by Philo Judaeus and Josephus, who inform us that the Jewish laws and customs were admired, imitated, and obeyed over the whole earth.334 Such assertions might be suspected of exaggeration, were they not, to a certain extent, confirmed by the references already quoted, to which others of the same kind may be added from later writers showing that it was a common practice among the Romans to abstain from work on the Sabbath, and even to celebrate it by praying, fasting, and lighting lamps, to visit the synagogues, to study the law of Moses, and to pay the yearly contribution of two drachmas to the temple at Jerusalem.335 Jeff’s hand was quietly coming down. “What happened to you?” begged Sandy. “Something new has come up, sir. I was waiting there by my ship a good while back, and I heard another one cruising and spiraling, shooting the field, I guess, because he came in and set down. My crate, just the way you ordered, was down by the grove, not in plain sight in the middle of the course. But Jeff set his ship down, left the engine running, and went off. I stayed hid to see what would happen, but when he didn’t come back, I thought I’d better go and find you—and see if it meant anything to you.” She stood alone, with the sticky, wet knife in her hand, catching her breath, coming out of the madness. Then she stooped, and pushing the branches aside felt about for her pistol. It lay at the root of a tree, and[Pg 80] when she had picked it up and put it back in the holster, there occurred to her for the first time the thought that the shot in the dead stillness must have roused the camp. And now she was sincerely frightened. If she were found here, it would be more than disagreeable for Landor. They must not find her. She started at a swift, long-limbed run, making a wide detour, to avoid the sentries, bending low, and flying silently among the bushes and across the shadowy sands. The year 1756 opened with menaces to England of the most serious nature. The imbecility of the Ministry was beginning to tell in the neglect of its colonies and its defences. France threatened to invade us, and a navy of fifty thousand men was suddenly voted, and an army of thirty-four thousand two hundred and sixty-three of native troops; but as these were not ready, it was agreed to bring over eight thousand Hessians and Hanoverians. To pay for all this it was necessary to grant excessive supplies, and lay on new duties and taxes. 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Si gave the boys the necessary instruction about having their things ready so that they could get them in the dark the next morning, and ordered them to disregard the bonfires and mirth-making, and lie down to get all the sleep they could, in preparation for the hard work of the next day. Then, like the rest of the experienced men, who saw that the campaign was at length really on, and this would be the last opportunity for an indefinite while to write, he sat down to write short letters to his mother and to Annabel. "Bully for the Wild Wanderers of the Wabash," Shorty joined in. "They're the boss regiment in the army o' the Cumberland, and the Army o' the Cumberland's the boss army on earth. Hooray for US Co. Le's have a speech. Where's Monty Scruggs?" "Bring a light, do—I can't abide this dark." Albert suddenly began to look uneasy. After all he was not really drunk, only a little fuddled. 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