Matrix Multiplication And Linear Transformations.Matrix multiplication is a fundamental operation in linear algebra that plays a crucial role in various fields of mathematics, science, and engineering. It’s closely connected to the concept of linear transformations.

## Matrix Multiplication And Linear Transformations

**Matrix Multiplication:** Given two matrices A and B, their product AB (read as “A times B”) is defined if and only if the number of columns in A is equal to the number of rows in B. If A is an m × n matrix (m rows and n columns), and B is an n × p matrix (n rows and p columns), then their product AB is an m × p matrix.

The elements of the resulting matrix AB are calculated using the dot product of rows from matrix A and columns from matrix B. Specifically, the entry in the i-th row and j-th column of the product matrix AB is obtained by taking the dot product of the i-th row of A and the j-th column of B.

**Linear Transformations:** A linear transformation (or linear map) is a function that maps vectors from one vector space to another, while preserving certain algebraic properties. In the context of matrix multiplication, linear transformations can be represented by matrices.

If we have a linear transformation T: V -> W, where V and W are vector spaces, and we choose bases for both vector spaces, then the linear transformation can be represented by a matrix. The matrix representation of a linear transformation depends on the choice of bases and is often referred to as the transformation matrix.

When you multiply a vector by a transformation matrix, you’re essentially applying that linear transformation to the vector. The resulting vector is the transformed version of the original vector under that linear transformation.

**Connection:** Matrix multiplication and linear transformations are interconnected in the sense that matrix multiplication can be used to perform and analyze linear transformations. The transformation matrix of a linear transformation represents how the transformation acts on the basis vectors of the domain space. Applying this matrix to other vectors will apply the same linear transformation to those vectors.

In summary, matrix multiplication allows us to represent and perform linear transformations efficiently and systematically, making it a powerful tool in various areas, including computer graphics, physics, engineering, and more.