Finding the Null Space of a Matrix: A Guide to Solving for the Solution Set

In summary: Remember that the nullspace is the set of all vectors that would send your original system to the zero vector.
  • #1
kkingkong
5
0

Homework Statement



What is the null space of this matrices.
|1 1 0 3 1|
|0 1 -1 0 1|
|1 1 3 0 1|


The Attempt at a Solution


I reduced it to rref using agumented matrice(each one equals to zero )
|1 0 0 4 0 0|
|0 1 0 -1 1 0|
|0 0 1 -1 0 0|

and i get get x1=-x4 , x2= x4 + -x5, x3 = x4
but then what is the null space??
 
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  • #2
kkingkong said:

Homework Statement



What is the null space of this matrices.
|1 1 0 3 1|
|0 1 -1 0 1|
|1 1 3 0 1|


The Attempt at a Solution


I reduced it to rref using agumented matrice(each one equals to zero )
|1 0 0 4 0 0|
|0 1 0 -1 1 0|
|0 0 1 -1 0 0|

and i get get x1=-x4 , x2= x4 + -x5, x3 = x4
but then what is the null space??
Your work so far is OK, but you just need to take it a little further.

Your final matrix, which BTW has an extra column (6th) that isn't needed, represents this system:
Code:
x[SUB]1[/SUB] = -x[SUB]4[/SUB]
x[SUB]2[/SUB] =  x[SUB]4[/SUB] - x[SUB]5[/SUB]
x[SUB]3[/SUB] =  x[SUB]4[/SUB]
x[SUB]4[/SUB] =  x[SUB]4[/SUB]
x[SUB]5[/SUB] =       x[SUB]5[/SUB]
If you sort of squint your eyes at what I wrote, you might be able to see that every vector x in the nullspace can be written as a linear combination of two vectors that are linearly independent.
 
  • #3
Remember that the nullspace is the set of all vectors that would send your original system to the zero vector.

If you could find a set of vectors that are independent and send your system to the zero vector, then it seems as though the span of those vectors would be your entire nullspace.

I think you can get it from there!
 
  • #4
Rellek said:
Remember that the nullspace is the set of all vectors that would send your original system to the zero vector.
Slight correction: the nullspace is the set of all vectors that your matrix would map to the zero vector.
Rellek said:
If you could find a set of vectors that are independent and send your system to the zero vector, then it seems as though the span of those vectors would be your entire nullspace.

I think you can get it from there!
 

1. What is the null space of a matrix?

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

2. How is the null space of a matrix calculated?

To calculate the null space of a matrix, we first reduce the matrix to row-echelon form. Then, the columns that contain the leading variables (corresponding to pivot positions in the row-echelon form) represent the basic variables, and the remaining columns represent the free variables. The null space can be expressed as a linear combination of the free variables.

3. What is the dimension of the null space?

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

4. What is the significance of the null space in linear algebra?

The null space has important applications in linear algebra, such as in solving systems of linear equations, finding the basis of a vector space, and determining the invertibility of a matrix. It also plays a crucial role in understanding the concept of linear independence and linear transformations.

5. Can the null space of a matrix be empty?

Yes, it is possible for the null space of a matrix to be empty. This occurs when the only solution to the equation Ax = 0 is the zero vector. In other words, when all the entries in the last row of the reduced row-echelon form of the matrix are zero, there are no free variables and thus no vectors in the null space.

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