Matrix multiplication by vector is a linear algebra operation that takes a matrix and a vector and returns another vector. The process involves multiplying each element in the matrix by the corresponding element in the vector and summing the products. The result is a new vector with the same number of rows as the matrix and one column.

## Matrix Multiplication By Vector with Examples

Let’s say we have a matrix A of size 3×2 and a vector X of size 2×1.

Matrix A: | a11 a12 |

| a21 a22 |

| a31 a32 |

Vector X: | x1 |

| x2 |

To multiply A and X, we will perform the following steps:

- Multiply the elements in the first row of A with the elements in X: a11 * x1 + a12 * x2
- Sum the products from step 1 to produce the first component of the resulting vector: result_1 = a11 * x1 + a12 * x2
- Repeat the process for each row in A: result_2 = a21 * x1 + a22 * x2

result_3 = a31 * x1 + a32 * x2

- The final result of the matrix-vector multiplication is a vector of size 3×1:

| result_1 |

| result_2 |

| result_3 |

This is the transformed version of the original vector X by matrix A.

Matrix-vector multiplication is used in various fields such as:

- Linear Algebra: Matrix-vector multiplication is a fundamental operation in linear algebra and is used to represent linear transformations, solve systems of linear equations, and eigenvalue problems.
- Computer Graphics: In 3D computer graphics, matrix-vector multiplication is used to rotate, translate, and scale objects in a virtual environment.
- Machine Learning: In machine learning, matrix-vector multiplication is used in the training and evaluation of neural networks. For example, it is used to compute dot products between weights and inputs in a neural network.
- Robotics: In robotics, matrix-vector multiplication is used to control robotic systems. Such as computing the end effector position from joint angles.
- Economics: In economics, matrix-vector multiplication is used to solve systems of linear equations representing production and utility functions in macroeconomic models.

#### Matrix-vector multiplication

The purpose of matrix-vector multiplication is to transform a vector by a matrix. This operation takes a matrix and a vector as inputs and returns a new vector. It representing the transformed version of the original vector. The resulting vector has the same number of rows as the matrix and one column.

The transformation performed by matrix-vector multiplication can be a linear transformation such as a rotation, a scaling, or a projection. The transformation is determined by the elements in the matrix. It can be used to represent various operations in fields such as linear algebra, computer graphics, machine learning, robotics, and economics.

In conclusion, matrix-vector multiplication is an important operation in linear algebra that transforms a vector by a matrix. The operation involves multiplying each element in the matrix by the corresponding element in the vector and summing the products. The result of the matrix-vector multiplication is a new vector with the same number of rows as the matrix and one column. It representing the transformed version of the original vector.