Question

How many rows and columns must a matrix A have in order to define a mapping from R⁴ into R⁵ by the rule T(x)=Ax?

A. 4 rows, 5 columns.
B. 5 rows, 4 columns.
C. 4 rows, 4 columns.
D. 5 rows, 5 columns.

Answer 1

In order to define a mapping from R^4 into R^5 by the rule T(x) = Ax, the matrix A must have 5 rows and 5 columns.

Step-by-step explanation:

In order to define a mapping from \(R^4\) into \(R^5\) by the rule \(T(x) = Ax\), the matrix \(A\) must have the same number of columns as the dimension of the input vector space and the same number of rows as the dimension of the output vector space.

Since \(x\) is a vector in \(R^4\), it has 4 components. Therefore, the matrix \(A\) must have 4 columns. Since the mapping is from \(R^4\) into \(R^5\), the output vector space has a dimension of 5. Therefore, the matrix \(A\) must have 5 rows. So, the correct answer is option D: 5 rows, 5 columns.