Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix AT whose columns are the rows of A. So: The columns of AT are the rows of A. The rows of AT are the columns of A. …

The transpose of a lower triangular matrix is upper triangular. The transpose of brings us back to . If we add two matrices and then transpose, the result is the same as first transposing and then adding: + …

Lecture 3c The Transpose of a Matrix (pages 120-121) As we start to explore the ways that a matrix is di erent from a giant . ector, we begin in what might seem. an unusual place. De nition: Let A be an m n …

The transpose is a more general concept than just an operation on matrices. Given a linear map T : X ! U, its transpose is a certain induced linear map T0: U0 ! X0 between the dual spaces. In the next …

To transpose a matrix, we re°ect it across the line given by the leading diagonal a11, a22 etc. In general the result is a di®erent shape to the original matrix:

W is a linear transformation and Tt is its transpose. Suppose A = f 1; :::; ng and. B = f 1; :::; mg are bases for V and W respectively and A0 = ff1; :::; fng and B0 = fg1; :::; gng are their dual bases for V …

The transpose of a matrix The transpose of an m n matrix A is the n m the entries of A across the main diagonal i = j: matrix A⊤; obtained by re ecting The relation between the entries is (A⊤)ij = Aji: If