Can You Switch Columns in a Matrix While Finding Its Null-Space?

[famallama]

Homework Statement

The actual problem is
Find a basis for the null-space of the matrix
(1 0 1 2 1)
(0 1 2 0 1)
(0 1 -1 3 1)

Homework Equations

there are no relevant equations.

The Attempt at a Solution

I attempted to get the matrix into RREF i got
(1 0 3 0 1)
(0 1 2 0 1)
(0 0 -1 1 0)
Each row consisting of a different variable x₁, x₂, x₃, x₄, x₅, respectively, I do not see an issue with using x₄ as a pivot and using x₃ as an open variable. Am i totally wrong in my assumption that the only thing that dictates where the variables are in the matrix are in fact their position in the original equation? so I switch the third and fourth column still calling them the same variable? Isn't this like if i were to have an equation x=2y+5g+j is the exact same equation as x=5g+j+2y? I ended up with the answer of
(-s-3t) (-3) (-1)
(-s-2t) (-2) (-1)
x=( t )=t(1 )+s(0 )
( t ) (1 ) (0 )
( s ) (0 ) (1 )
I stopped here because i didn't study right, but i got a 0 on it because i considered my RREF to be correct. Basically my biggest question is why am in not aloud to switch the columns in a matrix? I am aloud to switch the rows around if i please, but not being able to do the columns as well does not seem right to me.

 

[mighty2000]

it is important to understand that the operations you perform on a matrix must follow certain rules in order to maintain the integrity of the problem. In this case, you are trying to find a basis for the null-space of the matrix, which means finding a set of vectors that satisfy the equation Ax=0, where A is your given matrix.

When performing row operations on a matrix, you are essentially changing the equations that the matrix represents. However, when you switch columns in a matrix, you are essentially changing the variables that are being represented in the equations. This can lead to a different set of solutions and may not give you the correct basis for the null-space.

In your example, switching the third and fourth columns may have given you a different set of solutions, but it may not be the correct basis for the null-space. It is important to follow the established rules for matrix operations in order to arrive at the correct solution.

In summary, as a scientist, it is important to be precise and follow established rules when solving problems, especially in mathematics. Switching columns in a matrix may lead to a different set of solutions and may not give you the correct result. It is important to understand the implications of the operations you perform on a matrix and how they affect the problem at hand.