Nullspaces relation between components and overall matrix

In summary: So C*x = 0. Where x is the intersection point of N(A) and N(B), and that is the nullspace of C.In summary, the nullspace of matrix C is equal to the intersection of the nullspaces of matrices A and B. This is because, in order for C to produce a zero vector when multiplied by a vector x, both A and B must also produce zero vectors when multiplied by x. This means that x is in the nullspace of both A and B, making it the intersection of the two nullspaces.
  • #1
worryingchem
41
1

Homework Statement


If matrix ## C = \left[ {\begin{array}{c} A \\ B \ \end{array} } \right]## then how is N(C), the nullspace of C, related to N(A) and N(B)?

Homework Equations


Ax = 0; x = N(A)

The Attempt at a Solution


First, I thought that the relation between A and B with C is ## C = A + B ## so then I thought that ## N(C) = N(A) + N(B) ##.
But when I checked the solution it said N(C) = N(A) ∩ N(B)
and the only explanation is that ## Cx = \left[ {\begin{array}{c} Ax \\ Bx \ \end{array} } \right] = 0. ##
Can someone explain the solution to me?
 
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  • #2
worryingchem said:

Homework Statement


If matrix ## C = \left[ {\begin{array}{c} A \\ B \ \end{array} } \right]## then how is N(C), the nullspace of C, related to N(A) and N(B)?

Homework Equations


Ax = 0; x = N(A)

The Attempt at a Solution


First, I thought that the relation between A and B with C is ## C = A + B ## so then I thought that ## N(C) = N(A) + N(B) ##.
But when I checked the solution it said N(C) = N(A) ∩ N(B)
and the only explanation is that ## Cx = \left[ {\begin{array}{c} Ax \\ Bx \ \end{array} } \right] = 0. ##
Can someone explain the solution to me?

I'm not sure why you aren't getting the explanation. If ##\left[ \begin{array}{c} Ax \\ Bx \ \end{array} \right]## is the zero vector doesn't that mean that BOTH ##Ax## and ##Bx## must be zero vectors? Not just one or the other? ##C## isn't equal to ##A+B##, it's equal to ##\left[ \begin{array}{c} A\\ 0 \ \end{array} \right]+\left[ \begin{array}{c} 0 \\ B \ \end{array} \right]##. Even if it were the null space of ##A+B## is not equal to the null space of ##A## plus the null space of ##B##.
 
Last edited:
  • #3
I didn't understand how they went from ## Cx = \left[ {\begin{array}{c} Ax \\ Bx \ \end{array} } \right] = 0 ## to how N(C) is the intersection of N(A) and N(B). I just saw that the nullspaces were zero, but doesn't all nullspaces contain zero.

After some more thought, I did manage to visualize it if I use row vectors for ## A = \left[ {\begin{array}{cc} 1 & 1 \ \end{array} } \right] ## and ## B = \left[ {\begin{array}{cc} 1 & 2 \ \end{array} } \right] ##.
Then ## N(A) = \left[ {\begin{array}{c} 1 \\ -1 \ \end{array} } \right] ## and ## N(B) = \left[ {\begin{array}{c} 2 \\ -1 \ \end{array} } \right] ##. If I think of the nullspace as column vectors, then it's easier to visualize the nullspaces, and that they are perpendicular to the original matrix and intersect at 0.
When I make C, it would be ## \left[ {\begin{array}{cc} 1 & 1 \\ 1 & 2 \ \end{array} } \right] ## and ## N(C) = \left[ {\begin{array}{c} 0 & 0 \\ 0 & 0 \ \end{array} } \right] ##, the intersection of N(A) and N(B). Is this example right?

When I try to picture a plane though, I don't know how to define a plane in matrix notation and would the nullspace be a second plane with a normal vector orthogonal to the first plane's normal vector?
 
  • #4
worryingchem said:
I didn't understand how they went from ## Cx = \left[ {\begin{array}{c} Ax \\ Bx \ \end{array} } \right] = 0 ## to how N(C) is the intersection of N(A) and N(B). I just saw that the nullspaces were zero, but doesn't all nullspaces contain zero.

After some more thought, I did manage to visualize it if I use row vectors for ## A = \left[ {\begin{array}{cc} 1 & 1 \ \end{array} } \right] ## and ## B = \left[ {\begin{array}{cc} 1 & 2 \ \end{array} } \right] ##.
Then ## N(A) = \left[ {\begin{array}{c} 1 \\ -1 \ \end{array} } \right] ## and ## N(B) = \left[ {\begin{array}{c} 2 \\ -1 \ \end{array} } \right] ##. If I think of the nullspace as column vectors, then it's easier to visualize the nullspaces, and that they are perpendicular to the original matrix and intersect at 0.
When I make C, it would be ## \left[ {\begin{array}{cc} 1 & 1 \\ 1 & 2 \ \end{array} } \right] ## and ## N(C) = \left[ {\begin{array}{c} 0 & 0 \\ 0 & 0 \ \end{array} } \right] ##, the intersection of N(A) and N(B). Is this example right?

When I try to picture a plane though, I don't know how to define a plane in matrix notation and would the nullspace be a second plane with a normal vector orthogonal to the first plane's normal vector?

You don't have to visualize it that precisely. If ##x## is the nullspace of ##C##, then ##Cx=\left[ {\begin{array}{cc} Ax \\ Bx \ \end{array} } \right]=\left[ {\begin{array}{cc} 0 \\ 0 \ \end{array} } \right]##, right? Doesn't that mean ##Ax=0## and ##Bx=0##? And doesn't that mean the ##x## is in the nullspace of both ##A## and ##B##?
 
  • #5
Ah, I see.
Then, ## C*N(A) = \left[ {\begin{array}{c} A*N(A)\\ B*N(A) \ \end{array} } \right]=\left[ {\begin{array}{c} 0 \\ n \ \end{array} } \right] ##.
And ## C*N(B) = \left[ {\begin{array}{c} A*N(B)\\ B*N(B) \ \end{array} } \right]=\left[ {\begin{array}{c} n \\ 0 \ \end{array} } \right] ##.
n can be some non-zero numbers.
So in order to make ##\left[ {\begin{array}{c} 0 \\ 0 \ \end{array} } \right] ##, C has to be multiply by something that exists in both N(A) and N(B), the intersection point of the two, and that is x.
 

Related to Nullspaces relation between components and overall matrix

1. What is a nullspace in relation to components and overall matrix?

The nullspace of a matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In terms of components and overall matrix, the nullspace represents the parts of the overall matrix that have no effect on the components.

2. How do you calculate the nullspace of a matrix?

To calculate the nullspace of a matrix, you can use the reduced row echelon form (RREF) method or the nullity method. The RREF method involves performing row operations on the matrix until it is in its simplest form, and then identifying the columns with leading ones as the basis for the nullspace. The nullity method involves finding the basis for the nullspace by solving the homogeneous system of equations Ax=0.

3. What is the significance of the nullspace in matrix algebra?

The nullspace is significant because it represents the solutions to the homogeneous system of equations Ax=0. This means that it helps us understand the structure of the matrix and its relationship to its components. Additionally, the nullspace can be used to find the inverse of a matrix, which is important in solving systems of equations and other applications.

4. Can the nullspace of a matrix have more than one dimension?

Yes, the nullspace of a matrix can have more than one dimension. In fact, the dimension of the nullspace is equal to the number of free variables in the matrix. This means that the nullspace can have multiple vectors that span the same space, but they are all still considered part of the nullspace.

5. How is the nullspace related to the rank of a matrix?

The nullspace and the rank of a matrix are related through the rank-nullity theorem, which states that the nullity (dimension of the nullspace) plus the rank (dimension of the column space) of a matrix is equal to the number of columns in the matrix. This means that as the rank of a matrix increases, the dimension of the nullspace decreases, and vice versa.

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