Finding a Basis of the Null Space of a Matrix A in R^5 | SOLVED

To find a basis, we can use the Reduced Row Echelon Form of the matrix to find the pivot columns and then use those columns as the basis vectors.In summary, to find a basis of the null space N(A) of a matrix A, we can use the Reduced Row Echelon Form to find the pivot columns and use them as the basis vectors. The dimension of the null space can be determined by counting the number of pivot columns.
  • #1
karnten07
213
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[SOLVED] basis of a null space

Homework Statement



Find a basis of the null space N(A)[tex]\subset[/tex]R^5 of the matrix

A=
1 -2 2 3 -1
-3 6 -1 1 -7
2 -4 5 8 -4

[tex]\in[/tex]M3x5(R)

and hence determine its dimension

Homework Equations





The Attempt at a Solution



So do i need to find the x that satisfies Ax=0 and that x is the null space? Then how do i find a basis of this null space?
 
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  • #2
karnten07 said:
So do i need to find the x that satisfies Ax=0
Why do you think x is unique?

and that x is the null space?
No, the null space is the space of all solutions to the equation.
 

What is the null space of a matrix A in R^5?

The null space of a matrix A in R^5 is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In other words, it is the solution space of the homogeneous equation Ax=0.

Why is finding a basis of the null space important?

Finding a basis of the null space is important because it allows us to understand the structure of the solutions to the homogeneous equation Ax=0. It also helps us to determine the linear independence of the columns of A and the rank of the matrix.

How can the basis of the null space be found?

The basis of the null space can be found by first reducing the matrix A to its reduced row echelon form using elementary row operations. The columns corresponding to the leading variables will form the basis of the null space.

What is the dimension of the null space of a matrix A in R^5?

The dimension of the null space of a matrix A in R^5 is equal to the number of free variables in the reduced row echelon form of A. This can also be determined by subtracting the rank of A from the number of columns in A.

Can the basis of the null space of a matrix A in R^5 be empty?

Yes, it is possible for the basis of the null space to be empty. This would happen if the rank of A is equal to the number of columns in A, meaning there are no free variables in the reduced row echelon form and the homogeneous equation Ax=0 has only the trivial solution.

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