Question

Suppose a is a 7 times ×5 matrix. how many pivot columns must a have if its columns are linearly​ independent? why?

Answer 1

A matrix 7 * 5 means a matrix of 7 rows and 5 columns.
The matrix can only have 5 pivot columns for the system to be linearly independent.
Remember that in the system of equations:
Ax = b
A is the coefficient matrix of the incognita vector x and b is the solution vector.
The extended matrix is
(A l b)
If the matrix has more than 5 pivot columns, then the system is linearly dependent.