What is meant by transpose of a linear transformation explain with examples?

In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors.

Is transpose a conjugate?

It often happens in matrix algebra that we need to both transpose and take the complex conjugate of a matrix. The result of the sequential application of these two operations is called conjugate transpose (or Hermitian transpose). Special symbols are used in the mathematics literature to denote this double operation.

What is conjugate in linear algebra?

A conjugate matrix is a matrix obtained from a given matrix by taking the complex conjugate of each element of. (Courant and Hilbert 1989, p. 9), i.e., The notation. is sometimes also used, which can lead to confusion since this symbol is also used to denote the conjugate transpose.

How do you find the transpose of a linear transformation?

A linear transformation is a transformation between two vector spaces that preserves addition and scalar multiplication. Now if X and Y are two n by n matrices then XT+YT=(X+Y)T and if a is a scalar then (aX)T = a(XT) so transpose is linear on the n2 dimensional vector space of n by n matrices.

Is the transpose a linear operator?

The transpose of a scalar is the same scalar. Together with (2), this states that the transpose is a linear map from the space of m × n matrices to the space of all n × m matrices. The determinant of a square matrix is the same as the determinant of its transpose.

What is the conjugate transpose of a matrix called?

Other names for the conjugate transpose of a matrix are Hermitian conjugate, adjoint matrix or transjugate.

What is conjugate of a matrix with example?

Conjugate of a Matrix It is possible to find the conjugate for a given matrix by replacing each element of the matrix with its complex conjugate. Mathematically, a conjugate matrix is a matrix. A ― obtained by replacing the complex conjugate of all the elements of the matrix A.

Is the transpose operator?

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations).