Question

If A is a \({\bf{4}} \times {\bf{3}}\) matrix, what is the largest possible dimension of the row space of A? If Ais a \({\bf{3}} \times {\bf{4}}\) matrix, what is the largest possible dimension of the row space of A? Explain.

Short answer

If A is a \(4 \times 3\) matrix, then the largest possible dimension of the row space of A is 3.

If Ais a \(3 \times 4\) matrix, then the largest possible dimension of the row space of A is 3.

Step-by-step solution

Use the rank theorem

Note that the number of pivots in A gives the dimension of its column space.

By the rank theorem, \(\dim {\rm{Col}}\,A = \dim {\rm{Row}}\,A = {\rm{rank}}\,A\).

Compute the dimension of the row space for \({\bf{4}} \times {\bf{3}}\) matrix

If A is a \(4 \times 3\) matrix, then the number of pivots cannot exceed the number of columns. Here, the number of columns is minimum. Hence, the largest possible dimension of the row space of A is 3.

Compute the dimension of the row space for \({\bf{3}} \times {\bf{4}}\) matrix

If A is a \(3 \times 4\) matrix, then the number of pivots cannot exceed the number of rows. Here, the number of rows is minimum. Hence, the largest possible dimension of the row space of A is 3.