Question

How many pivot columns must a \(5 \times 7\) matrix have if its columns span \({\mathbb{R}^5}\)? Why?

Short answer

The \(5 \times 7\) matrix must have five pivot columns.

Step-by-step solution

Determine the pivot position if the columns span \({\mathbb{R}^5}\)

Theorem 4 states that\(A\)is an\({\mathop{\rm m}\nolimits} \times n\)matrix. For this, the following statements are equivalent.

1. For each \({\mathop{\rm b}\nolimits} \) in \({\mathbb{R}^m}\), the equation \(Ax = b\) has a solution.
2. Each \({\mathop{\rm b}\nolimits} \) in \({\mathbb{R}^m}\) is a linear combination of the columns of A.
3. The columns of \(A\) span .
4. \(A\)has a pivot position in every row.

According to theorem 4, if the columns of a \(5 \times 7\) matrix \(A\) span \({\mathbb{R}^5}\), matrix \(A\) has a pivot in every row.

Determine the number of pivot columns in the \(5 \times 7\) matrix

Matrix \(A\) contains five pivot columns since each pivot position is in a different column.

Thus, the \(5 \times 7\) matrix must have five pivot columns.