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Chapter 1: Q27E (page 1)

How many pivot columns must a \(7 \times 5\) matrix have if its columns are linearly independent? why?

Short Answer

Expert verified

The matrix must have five pivot columns so that the columns are linearly independent.

Step by step solution

01

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

If the columns of a \(7 \times 5\) matrix are linearly independent, the matrix must have five pivot columns.

02

Describe why the \(7 \times 5\) matrix has five pivot columns

The column of matrix \(A\) islinearly independent if and only if the equation \(Ax = 0\) has only a trivial solution.

The matrix must contain five pivot columns if the columns are linearly independent.

It is because the equation \(Ax = 0\) must have only a trivial solution. Alternatively, the equation \(Ax = 0\) can have a free variable for the columns of \(A\) to be linearly dependent.

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

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