# Semantics of Views

#### Subranging

Subranging takes a number of range restrictions and produces a matrix view which has the same number of dimensions but different shape. For example, restricting the range to the last 5 indexes in each dimension again produces a 3-dimensional matrix (view) but now with less extent.

#### Slicing

Slicing blends out one or more dimensions. It produces a matrix view which is lower dimensional than the original. In the above picture, the second dimension has been fixed to index 2, yielding a flat two-dimensional plate. Since the view has a 2-dimensional type it will accept any operation defined on two-dimensional matrices and may be used as argument to any external method operating on 2-dimensional matrices.

#### Dicing

Dicing virtually rotates the matrix. It exchanges one or more axes of the coordinate system. Thus, a 3 x 4 matrix can be seen as a 4 x 3 matrix, a 3 x 4 x 5 matrix can be seen as a 5 x 3 x 4 matrix, and so on. Dicing produces a view with the same dimensionality but different shape.

#### Flipping

Flipping mirrors coordinate systems. What used to be the first index becomes the last, ..., what used to be the last index becomes the first. Thus, a matrix can be seen from the "left", the "right", the "top", the "bottom", the "front", the "backside", etc. Flipping produces a view with the same dimensionality and the same shape.

#### Striding

Striding blends out all but every i-th cell. It produces a view with the same dimensionality but smaller (or equal) shape.

#### Selecting

Selecting blends out all but certain indexes of slices, rows, columns. Indexes may have arbitrary order and can occur multiple times. Selecting produces a view with the same dimensionality but different shape (either larger or smaller).

#### Sorting

Sorting reorders cells along one given dimension. It produces a view with the same dimensionality and the same shape but different cell order.

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#### Combinations

All views are orthogonal to each other. They can be powerful tools, particularly when applied in combination. Feeding the result of one view transformation into another transformation can do complex things.

#### Copying, Assignment & Equality

Any matrix and view can be copied. Copying yields a new matrix equal to the original (view) but entirely independent of the original. So changes in the copy are not reflected in the original, and vice-versa.
Two matrices are equal if they have the same dimensionality (rank), value type, shape and identical values in corresponding cells.
Assignment copies the cell values of one matrix into another matrix. Both matrices must have the same dimensionality and shape.