Understanding Matrix Solutions: AX=B vs. AX=0

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In summary, the conversation discusses the properties of a matrix and its row echelon form. It is stated that if the number of rows is greater than the number of steps in the row echelon form, there exists a column in which the set of equations has no solutions. If the number of columns is greater than the number of steps, the matrix equation AX=0 has a solution expressed parametrically by (n-d) parameters. And if the number of rows, columns, and steps are equal, there exists a unique solution for every B in the equation AX = B. The conversation also delves into the concept of nullspace and how it relates to the number of parameters in the solution. Finally, it is mentioned that using standard
  • #1
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Let A denote an mxn matrix and let A' denote the row echelon form of it, which has d steps. We then have according to my textbook:
1) If m>d there exists a column such that the set of equation has no solutions.
2) If n>d the matrixequation AX=0 has a set of solution expressed parametrically by (n-d) parameters.
3) If m=n=d there exists a unique solution for every B in the equation AX = B.

Now 1) and 3) I understand. What troubles me is 2). Why to they switch the equation to AX=0 rather than AX=B? Wouldn't that last equation also be dependent on n-d parameters. I'm pretty sure that this has something to do with the fact that the nullspace is quite a unique thing since it forms a linear subspace. Thus we can define its dimension and later use all this to prove the rank nullity theorem. So can someone explain what's going on on a deeper level?
 
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  • #2
I assume by steps you mean pivots. In this case I prefer the term "rank" i.e. rank = d.Solution for problem AX = b is X = X_p + X_n.
Where,

AX = A(X_p) + A(X_n) = b + 0 = b.

X_p = particular solution
X_n = solution of Ax = 0 ... (n is not a number here, it is just a symbolic name)(n-d) parameters are the free variables where (n-d) is the dimension of the null space.

What this means is that you have a whole space of dimension (n-d) and each vector in that space is a solution of Ax=0. Hence the name nullspace.
 
  • #3
You use the word "column" in your first claim. By column do you mean the column vector b of Ax = b or a column of matrix A?

You ought to use standard terms here. If the book you're reading is using "column" for "column vector" and "steps" for "pivots", I suggest that you throw it away.
 
  • #4
in part 2, you could use AX=B, but the statement would be more complicated. I.e. you could say, for every B, AX=B either has no solutions or has solutions depending on n-d parameters. but in fact this is implied by the special case AX=B, since if C is a solution of AC=0, and if D is a solution of AD=B, then also A(C+D) = B. So as long as AX=B has at least one solution, then its set of solutions is in one one correspondence with the solutions of AX=0. (I.e. notice that A0=0 is true, so when B=0, the equation Ax=B does always have at least one solution.)
 
  • #5


The difference between AX=B and AX=0 lies in the fact that the first equation represents a system of linear equations with a non-zero right hand side, while the second equation represents a homogeneous system of linear equations with a zero right hand side.

In the case of AX=B, the solution set represents all possible values of the unknown variables that satisfy the given equations. However, in the case of AX=0, the solution set represents a special case where all the unknown variables are equal to zero. This special case is known as the nullspace or kernel of the matrix A.

When m>d, it means that there are more equations than unknowns, which results in an overdetermined system. In this case, there may not be a unique solution or there may be no solution at all. However, when n>d, it means that there are more unknowns than equations, resulting in an underdetermined system. In this case, there may be infinitely many solutions, which can be expressed parametrically by (n-d) parameters.

The reason why the equation is switched to AX=0 when discussing the nullspace is because the nullspace is defined as the set of all solutions to the homogeneous equation AX=0. This is because any vector in the nullspace, when multiplied by A, will result in a zero vector, satisfying the equation AX=0. Therefore, the nullspace is a special solution set that is only relevant when the right hand side is zero.

The concept of nullspace is important in linear algebra as it helps us understand the relationship between the dimensions of the row and column spaces of a matrix. The rank-nullity theorem states that the dimension of the nullspace plus the dimension of the column space equals the number of columns in the matrix, which is equal to d in this case. Therefore, by studying the nullspace, we can gain insight into the rank and nullity of a matrix, which has many applications in mathematics and other fields such as engineering and physics.

In summary, the switch from AX=B to AX=0 is necessary when discussing the nullspace, as it represents a special case of the solution set where all unknown variables are equal to zero. This helps us understand the relationship between the dimensions of the row and column spaces of a matrix, and is essential in proving the rank-nullity theorem.
 

1. What is the difference between AX=B and AX=0 in matrix solutions?

AX=B represents a system of linear equations where the solution is a specific set of values for the variables in the equation. On the other hand, AX=0 represents a homogeneous system of linear equations where the only solution is when all variables are equal to 0.

2. Can AX=B and AX=0 have the same solution?

Yes, it is possible for both equations to have the same solution. This can happen when the values of the variables in AX=B are all equal to 0, making it the same as the solution for AX=0.

3. How do I solve for the solution in AX=0?

The solution for AX=0 is known as the trivial solution, where all variables in the equation are equal to 0. This can be solved by using row operations to reduce the matrix to its row-echelon form, revealing the values of the variables.

4. Is it possible for AX=B to have no solution?

Yes, it is possible for AX=B to have no solution. This happens when the system of equations is inconsistent, meaning there is no set of values for the variables that satisfy all equations simultaneously.

5. Can I use the same method to solve AX=B and AX=0?

No, the methods for solving these two types of equations are different. AX=B can be solved using elimination or substitution methods, while AX=0 can only be solved using row operations to reduce the matrix to its row-echelon form.

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