Question

Let A be a \(6 \times 5\) matrix. What must a and b in order to define \(T:{\mathbb{R}^{\bf{a}}} \to {\mathbb{R}^{\bf{b}}}\) by \(T\left( x \right) = Ax\)?

Short answer

The values must be \(a = 5\) and \(b = 6\).

Step-by-step solution

Write the condition for the transformation of dimension

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 in the codomain of transformation \(T\).

Observe the domain and codomain of transformation T

The given transformation is \(T:{\mathbb{R}^a} \to {\mathbb{R}^b}\).

Thus, \({\mathbb{R}^a}\) is the domain and \({\mathbb{R}^b}\) is the codomain.

Obtain the values of a and b

It is given that the order of the matrix is\(6 \times 5\). It means there are six rows and five columns.

Here,\({\mathbb{R}^5}\)is the domain and\({\mathbb{R}^6}\)is the codomain of transformation\(T\).

Thus, \(a = 5\) and \(b = 6\).