Transformation matrix
An image array or matrix is a matrix representation that is used in linear algebra, to describe a linear transformation between two finite-dimensional vector spaces.
The derived from these affine transformations, and Affinities projectivities can also be represented by imaging matrices.
- 2.1 pictures on coordinate tuple
- 2.2 Illustrations in general vector spaces
- 5.1 change of basis
- 5.2 Description of endomorphisms
- 5.3 Description of affine transformations, and Affinities
- 6.1 orthogonal projection
- 6.2 Reflection
- 6.3 rotation
Term
Requirements
In order to describe a linear map of vector spaces by a matrix, a base must first have been (with the order of the basis vectors ) is chosen fixed both in the prototype space and the target space. With a change of bases in one of the rooms in question, the matrix must be transformed, otherwise it describes a different linear mapping.
If in the definition and quantity of the target set a base has been selected, a linear mapping can be uniquely described by a mapping matrix. However, it must be set for whether you record the coordinates of vectors in column or row notation. The more common spelling is in columns.
Therefore one must the vector to be mapped as a column vector (with respect to the chosen basis) write.
Structure when using column vectors
After selecting a base from the definition of the quantity and the target amount are in the columns of the image matrix, the coordinates of the images of the basis vectors of the vector space depicted relative to the base of the target space: Each column of the matrix is the image of a vector of the prototype base. A projection matrix that describes a mapping from a 4-dimensional vector space into a 6-dimensional vector space, so it should always 6 rows ( for the six image coordinates of the basis vectors ) and 4 columns (one for each basis vector of the original image space, a ) have.
Generally, a linear imaging array of a n-dimensional vector space into an m-dimensional vector space having m rows and n columns. The image of a coordinate vector can then be calculated as:
Here is the image vector, the vector is mapped in each of the members of the chosen based on their space coordinates.
See also: Structure of the mapping matrix.
Use of row vectors
If, instead of column - row vectors, then the projection matrix must be transposed. This means that now the coordinates of the image of the first basis vector in the original image space in the first row are, etc. In calculating the coordinates of the image ( row coordinate ) vector must now be multiplied from the left on the picture matrix.
Calculation
Pictures on coordinate tuple
Let be a linear map and
An ordered basis of
As a basis for the target amount, the default base is chosen:
The projection matrix is obtained by conceives the images of the basis vectors of V as columns of a matrix:
Example: Consider the linear map
Both in the original image space and the target space, the default base is chosen:
The following applies:
Thus, the projection matrix is chosen with respect to the bases and:
Pictures in general vector spaces
If the elements of the target area are not coordinate tuple, or other reasons, a different base is selected in place of the standard basis, the images to be represented as a linear combination of base vectors to determine the entries of the transformation matrix:
The projection matrix is then obtained by column-wise entering the coefficients of the linear combinations in the matrix:
Example: It will again consider the linear mapping of the above example. This time, however, is in the target area, the ordered basis
Considered. Now, the following applies:
Thus one obtains for imaging matrix of with respect to the bases and:
Coordinate representation of linear maps
Using the transformation matrix can be calculated the image vector of a vector of the linear mapping.
If the vector relative to the base the coordinate vector
That is
And has the image vector with respect to the basis of the coordinates
That is
So true
Or with the help of the projection matrix expressed:
Short
Or
Sequential execution of linear maps
The sequential execution of linear maps corresponds to the matrix product of the associated imaging matrices:
There were, and vector spaces over the field and and linear maps. In the higher-level base is added to the base and the base. Then we obtain the projection matrix of the concatenated linear mapping
By the projection matrix and of the matrix of Figure (each relative to the corresponding bases) multiplying:
Note that must be selected in the same for both imaging matrices basis.
Justification: It is, and. The th column contains the coordinates of the image of the -th basis vector of respect to the base:
If we calculate the right hand side using the imaging matrices of and, we obtain:
By comparing coefficients follows
For all and, hence,
That is:
Use
Change of basis
If the projection matrix of an image for certain bases are known, then leaves the projection matrix for the same figure, however, is easy to calculate with other bases. This process is referred to as the base change. It may mean that the present bases are poorly suited to solve a particular problem with the matrix. By a base change, the matrix is then present in a simpler way, but still represents the same linear image. The mapping matrix is calculated from the projection matrix and the Basiswechselmatrizen and as follows:
Description of endomorphisms
In a linear self-map ( a linear operator ) of a vector space is usually lays a firm basis of the vector space underlying a set of definitions and target quantity. Then, the mapping matrix describes the variation experienced by the coordinates of an arbitrary vector with respect to the base in the figure. The mapping matrix is always square with endomorphisms, ie the number of lines matches the number of columns.
Description of affine mappings and affinities
After the election of an affine point basis in both affine spaces, which are mapped by an affine transformation to each other, this illustration may be through an imaging matrix and an additional shift or - in homogeneous coordinates by an extended: describe (or " homogeneous " ) Figure matrix alone.
Examples
Orthogonal projection
In three-dimensional space ( with the canonical basis ), one can describe the orthogonal projection of a vector on a line through the origin by the following transformation matrix:
Here, the coordinates of the normalized direction vector of the straight line. If instead of a straight line on a level with the two mutually perpendicular, normalized direction vectors and projected, one can interpret this in two projections along the two direction vectors, and thus set up the projection matrix for the orthogonal projection on a plane origin as follows:
The projection matrix to project onto a plane, so the sum of projection matrices on their direction vectors.
Reflection
A mirror carried out instead of a projection, it can also be shown using the above projection matrix. For the reflection matrix on a line through the origin with normalized directional vector holds:
The unit matrix represents. The same applies to the reflection in the plane:
Rotation
When you turn to a line through the origin with normalized direction vector in three-dimensional space, the necessary for this rotation matrix is represented as follows:
Wherein again denotes the unit matrix and the rotation angle.