Finding null space of a given matrix

In summary: Any point (x, y, z) that satisfies that equation is a point in the null space, NS(A), of A. If you are to "sketch NS(A) in R^2 or R^3", you need to specify which two or three variables you want to represent on the axes. For example, if you want to represent the null space as a curve in the xz-plane, solving for y gives y= (-x- 2z)/3. The graph of that curve will be the null space of A.
  • #1
Maxwhale
35
0

Homework Statement



Find null space of A, NS(A) and sketch NS(A) in R2 or R3.

A = [1 3 2; 2 6 4]

Homework Equations



AX = 0


The Attempt at a Solution



I know the second row is twice the first one. I tried to solve for x1, x2 and x3 putting everything in the form of AX =0. I did not get a confident answer to sketch.
 
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  • #2
What did you get?
 
  • #3
since there are three variables, x1, x2 and x3, i reduced them to row echelon presuming that x3=0 and then got x1=0 and x3=0. Is that a right way?
 
  • #4
What do you mean? The null space of A is a set.
 
  • #5
Either say NS(A) is the set of vectors (x, y, z) such that ... or, perhaps simpler, give a basis for the Null Space.
 
  • #6
We are not that far yet. We have not covered basis. I would really appreciate if you could put it simply so that i could understand. Please :)
 
  • #7
Maxwhale said:

Homework Statement



Find null space of A, NS(A) and sketch NS(A) in R2 or R3.

A = [1 3 2; 2 6 4]

Homework Equations



AX = 0


The Attempt at a Solution



I know the second row is twice the first one. I tried to solve for x1, x2 and x3 putting everything in the form of AX =0. I did not get a confident answer to sketch.

Maxwhale said:
since there are three variables, x1, x2 and x3, i reduced them to row echelon presuming that x3=0 and then got x1=0 and x3=0. Is that a right way?
You cannot assume x3 equals any specific number. The nullspace of A is all vectors <x1, x2, x3> such that
[tex]\left[\begin{array}{ccc} 1 & 3 & 2 \\ 2 & 6 & 4 \end{array}\right]\left[\begin{array}{c} x1 \\ x2 \\ x3\end{array}\right]= \left[\begin{array}{c}0 \\ 0 \\ 0 \end{array}\right][/tex]

You can row reduce that as you said you did: the bottom row will be 0, of course, the top row [1 3 2 0] which corresponds to the equation x1+ 3X2+ 2X3= 0. Since that is single equation in 3 unknown numbers, you can choose two of them to be whatever you want and solve for the third. If x2= 1 and x3= 0, what is x1?. If x2= 0 and x3= 1, what is x1?

What is the dimension of NP(A)? What is a basis for NP(A).
You can,
 
  • #8
The matrix A which you gave has one pivot column making the other two columns free.

n-r = 3-1 = 2, the dimension of the nullspace is 2.

Set the free variables to 1 and 0, and use back substitution to find the missing variable in your nullspace vector. You should get:

s_1 = [-3, 1, 0] and s_2 = [-2, 0, 1], with dimension 2.

I'm pretty sure that's right.
 
  • #9
FourierX said:
We are not that far yet. We have not covered basis. I would really appreciate if you could put it simply so that i could understand. Please :)
If you do not know what a basis is then I guess that by "find the null space" you are meant to just write the equation satisfied by points in the null space. As you said, the two equations are equivalent so any such point must satisfy x+ 3y+ 2z= 0.

What is the graph of that?
 

1. What is the null space of a matrix?

The null space of a matrix, also known as the kernel, is the set of all vectors that, when multiplied by the matrix, result in the zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the given matrix.

2. How do you find the null space of a matrix?

To find the null space of a matrix, you can use row reduction techniques to convert the matrix into reduced row echelon form. The columns without a pivot position (leading 1) will form the basis for the null space. You can then express the null space as a linear combination of these basis vectors.

3. What is the dimension of the null space?

The dimension of the null space is equal to the number of columns in the matrix minus the rank of the matrix. This can also be represented as the number of free variables in the reduced row echelon form of the matrix.

4. Can a matrix have more than one null space?

No, a matrix can only have one null space. However, the null space can have multiple basis vectors which span the same subspace.

5. How is the null space related to the column space?

The null space and the column space of a matrix are complementary subspaces. In other words, the vectors in the null space are orthogonal to the vectors in the column space. This means that the dot product of any vector in the null space and any vector in the column space will always be equal to zero.

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