This applet illustrates the concept of first derivative, as the best of a series of succesive approximations to the slope in a point.

## Monday, March 26, 2012

## Friday, March 23, 2012

### Poisson distribution

This applet represents some features about the Poisson distribution, very useful to describe the probability of events wich occur with a known average rate and independtly of the occurrence of the last event, like nuclear decay processes or even car crashes.

## Sunday, March 18, 2012

### Polar coordinates

This applet plots a function specified in polar coordinates, being x the angle in radians.

## Wednesday, March 14, 2012

### Eigenvalues and eigenvectors

This applet is based in the one on Linear mappings, being essentially the same with the option of displaying eigenvectors and eigenvalues.

The black vector represents the input of the linear application, it can be edited by click and drag. The red vector represents the result of the mapping. The mapping matrix can also be edited. Remember that not all linear mappings have real eigenvalues.

Questions:

· What happens if the black vector lies over an eigendirection? Is there any relation with the eigenvalues?

· Introduce the following matrix a = 0, b = 1, c = -1, d = 0. Move the vector and try to guess what the transformation represents. It has no real eigenvectors, why?

· Is it possible for a two dimensional linear mapping to have more than two eigenvectors? Why?

· What happens if we transform the application to a basis of eigenvectors?

The black vector represents the input of the linear application, it can be edited by click and drag. The red vector represents the result of the mapping. The mapping matrix can also be edited. Remember that not all linear mappings have real eigenvalues.

Questions:

· What happens if the black vector lies over an eigendirection? Is there any relation with the eigenvalues?

· Introduce the following matrix a = 0, b = 1, c = -1, d = 0. Move the vector and try to guess what the transformation represents. It has no real eigenvectors, why?

· Is it possible for a two dimensional linear mapping to have more than two eigenvectors? Why?

· What happens if we transform the application to a basis of eigenvectors?

## Monday, March 12, 2012

### Coordinates

This applet aids to visualize the concept of coordinate and the concept of change of basis in linear algebra.

The red vector is editable by click and drag, and so are the basis vectors.

Observe that not only the orientation of the new base vectors is important, but also the module.

Try to insert two linearly dependent vectors as generators of the base B. You'll get an error message. Why?

The red vector is editable by click and drag, and so are the basis vectors.

Observe that not only the orientation of the new base vectors is important, but also the module.

Try to insert two linearly dependent vectors as generators of the base B. You'll get an error message. Why?

### Taylor series

This applet shows the first 5 Taylor polynomials of a given function. The function is editable, and also the center of the neighborhood.

Observe how the successive orders converge to the given functions.

What happens if the function is a polynomial of order 5 or less?

What happens if the function has discontinuities, like tan(x)?

What happens if the function is not differentiable, like abs(x)?

Observe how the successive orders converge to the given functions.

What happens if the function is a polynomial of order 5 or less?

What happens if the function has discontinuities, like tan(x)?

What happens if the function is not differentiable, like abs(x)?

## Sunday, March 11, 2012

### Binomial distribution

The binomial distribution describes the probability of obtaining some number of successes in a sequence of n independent yes/no experiments. For example, the probability of obtaining exactly 3 heads in 5 rounds of coin tossing is P(5;3,0.5), being 0.5 the probability of success in a single round.

## Saturday, March 10, 2012

### Linear mappings

This applet shows the behaviour of a linear mapping in two dimensions. The image vector is shown in red, and the black vector can be displaced by click and drag.

The transformation matrix is editable.

Questions:

1. Construct the image of the vector (0,0). What happens?

2. Construct the image of the vector (1,1), and afther that, the image of the vector (2,2). What happens? Why?

3. Can you estimate, just playing with the vector, some eigenvector? Remember that not all linear mappings have real eigenvectors.

Related applets:

Eigenvalues and eigenvectors.

The transformation matrix is editable.

Questions:

1. Construct the image of the vector (0,0). What happens?

2. Construct the image of the vector (1,1), and afther that, the image of the vector (2,2). What happens? Why?

3. Can you estimate, just playing with the vector, some eigenvector? Remember that not all linear mappings have real eigenvectors.

Related applets:

Eigenvalues and eigenvectors.

### Sines and cosines

This applet allows you to visualize the sine and cosine of an angle inside a goniometric circumference.

Subscribe to:
Posts (Atom)