Question

How many rows and columns must a matrix a have in order to define a mapping from into by the rule t(x) = ax?

Answer 1

To define a mapping using the rule t(x) = ax, the matrix a must have m rows and n columns.

Step-by-step explanation:

To define a mapping from n-dimensional space into m-dimensional space using the rule t(x) = ax, the matrix a must-have m rows and n columns.

Each column in the matrix a represents the coefficients of the linear transformation for each dimension in the input space, while each row represents the transformation applied to each dimension in the output space.

For example, if we have a mapping from 2-dimensional space into 3-dimensional space, the matrix a would have 3 rows and 2 columns. This matrix configuration underscores the systematic correspondence between dimensions, pivotal in multivariate transformations and diverse mathematical applications.