Matriisivektoritulojen

Matriisivektoritulojen, also known as matrix-vector multiplication, is a fundamental operation in linear algebra. It involves multiplying a matrix by a vector. The result of this operation is another vector. For matrix-vector multiplication to be defined, the number of columns in the matrix must equal the number of elements in the vector.

Let A be an m x n matrix and v be an n x 1 column vector. The product Av is an m x 1 column vector, let's call it w. Each element of w is calculated as the dot product of a row of A and the vector v. Specifically, the i-th element of w, denoted as w_i, is given by the sum of the products of the elements of the i-th row of A and the corresponding elements of v. Mathematically, w_i = sum(A_{ij} * v_j) for j from 1 to n.

Matrix-vector multiplication has numerous applications across various fields. In computer graphics, it is used to transform points and vectors, such as scaling, rotation, and translation. In machine learning, it forms the basis of many algorithms, including neural networks, where it is used to compute the output of layers. It is also prevalent in solving systems of linear equations, where a system can be represented in the form Av = b, and matrix-vector multiplication is a key step in iterative solution methods. The interpretation of matrix-vector multiplication as a linear transformation is also crucial, where the matrix acts on the vector to produce a transformed vector.