Vänstermultiplikation

Vänstermultiplikation is a mathematical operation involving matrices. When multiplying a matrix A by a matrix B on the left, the result is denoted as AB. This operation is only defined if the number of columns in matrix A is equal to the number of rows in matrix B. If A is an m x n matrix and B is an n x p matrix, then the resulting matrix AB will have dimensions m x p.

The process of vänstermultiplikation involves calculating each element of the resulting matrix. The element in the i-th row and j-th column of AB is found by taking the dot product of the i-th row of A and the j-th column of B. This means summing the products of the corresponding elements from the row and the column.

Vänstermultiplikation is not commutative, meaning that in general, AB is not equal to BA. The order of multiplication is crucial. The dimensions must also align for the operation to be possible. For example, if A is 2x3 and B is 3x4, then AB is defined and results in a 2x4 matrix. However, BA would not be defined because the number of columns in B (4) does not equal the number of rows in A (2).

This operation is fundamental in linear algebra and has applications in various fields, including computer graphics, physics, and economics, often used to represent transformations or systems of equations.