Linear transformations in 3D

  • Linear transformations in 3D occur when all vector lines remain straight and uniform following transformation.

  • Linear transformations involve two properties: vector addition and scalar multiplication. The transformed sum of two vectors equals the sum of their independent transformations and the transformed product of a vector and a scalar equals the product of the scalar and the transformed vector.

  • The rotation, scaling, and reflection of vectors in 3D space are examples of linear transformations.

  • The exact transformation a vector undergoes can be represented with a matrix. This matrix representation makes it easy to calculate the results of combining multiple transformations.

  • Any 3D linear transformation can be expressed as a 3x3 matrix with the entries representing the transformed components of each of the basis vectors i, j, and k.

  • In the 3D matrix, the first row represents the new values of the i vector, the second row the j vector and the third row the k vector following the transformation.

  • The determinant of a transformation matrix can be used to understand the type of transformation. If it’s equal to 1, the transformation is isometric meaning it preserves lengths and angles. If it’s not equal to 1, the transformation includes some scaling.

  • Shearing is another type of linear transformation. In 3D, it involves changing one or two coordinates while keeping the rest fixed.

  • Linear transformations preserve various properties of vectors, such as lines and vector relationships. For this reason, the properties of vectors, such as addition and scalar multiplication, do not change under linear transformation.

  • The inverse transformation is a transformation that restores any transformations performed on a vector. It can be calculated using the inverse of the transformation matrix provided the matrix is not singular (i.e., its determinant does not equal zero).

  • Composing transformations involves performing one transformation after another. In matrix terms, this is achieved by multiplying the transformation matrices.

  • Eigenvalues and eigenvectors are key concepts in linear transformations. An eigenvector of a transformation is a non-zero vector that changes only by a scalar factor (the eigenvalue) under that transformation.