Non-pivot columns are linear combinations of the pivot columns.

[HelpNeeder]

Nonpivot-columns of a matrix A are linear combinations of the pivot-columns of A.

Could someone explain to me why this is true? Thank you.

 

[Deleted member 4993]

Nonpivot-columns of a matrix A are linear combinations of the pivot-columns of A.

Could someone explain to me why this is true? Thank you.

Please tell us the definition of a

​

Pivot Column of a matrix.​

 

[HelpNeeder]

If a matrix is in row-echelon form, then the first nonzero entry of each row is called a pivot, and the columns in which pivots appear are called pivot columns.

 

[Steven G]

That is the correct definition.
Now why do some columns not have pivots? Think hard about this. Look at concrete examples and see if the columns that do not have a pivot in them are in fact linear combinations of the columns that do have pivots in them. After looking at a few examples think about why this happens.

Please post back showing us the work you did for a couple of matrices like I asked for above. Include your reason why you think this happens. If you do not see why, then we'll give you another hint.