The use of subscript and superscript in linear algebra notation is to refer to specific elements or indices in a matrix or vector. Subscripts are used to indicate the row and column number of an element, while superscripts are used to indicate the power or exponent of a number.
A matrix is a rectangular array of numbers, while a vector is a one-dimensional array of numbers. Matrices are typically used to represent linear transformations, while vectors are used to represent quantities with both magnitude and direction.
The dot product is used to measure the similarity between two vectors and is represented using the symbol "⋅" to differentiate it from the cross product, which is represented using the symbol "×". This notation was introduced by Oliver Heaviside in the late 19th century.
Matrix multiplication is represented using the symbol "⋅" or simply by placing two matrices next to each other. For example, A⋅B or AB represents the multiplication of matrix A and matrix B. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
The transpose of a matrix is represented using the symbol "T" and it is used to switch the rows and columns of a matrix. This operation is useful in many applications, such as solving systems of linear equations and finding the inverse of a matrix.