How many rows and columns must a matrix A have in order to define a mapping from \({\mathbb{R}^4}\) into \({\mathbb{R}^5}\) by the rule \(T\left( x \right) = Ax\)?

For a matrix of order\(m \times n\), if vector\({\bf{x}}\)is in\({\mathbb{R}^n}\), then transformation\(T\)of vector x is represented as\(T\left( x \right)\)and it is in the dimension\({\mathbb{R}^m}\).

It can also be written as \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\).

Here, the dimension \({\mathbb{R}^n}\) is the domain, and the dimension \({\mathbb{R}^m}\) is the codomain of transformation \(T\).

Since the transformation is from \({\mathbb{R}^4}\) to \({\mathbb{R}^5}\), or \(T:{\mathbb{R}^4} \to {\mathbb{R}^5}\), \({\mathbb{R}^4}\) is the domain and \({\mathbb{R}^5}\) is the codomain.

Assume that the order of the matrix is\(m \times n\). It means there are\(m\)rows and\(n\)columns.

Since\({\mathbb{R}^4}\)is the domain and \({\mathbb{R}^5}\) is the codomain of the transformation, \(n = 4\)and\(m = 5\).

Therefore, matrix Amust have five rows and four columns to define the mapping.