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A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
Composition has closure and is associative, since matrix multiplication is associative. Notice that both Ref(θ) and Rot(θ) have been represented with orthogonal matrices. These matrices all have a determinant whose absolute value is unity. Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1.
A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder. [1]
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as , as a linear transformation of the space , the matrices and represent rotations or reflection of the space, while represents the scaling of each coordinate by the factor .
A quick way to model this reflection is with the method of images. The reflections, or images, are oriented in space such that they perfectly replace any mass (from the real plume) passing through a given boundary. [3] A single boundary will necessitate a single image. Two or more boundaries produce infinite images.
Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite ...
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