Null space

Null space

Lessons

The null space of a matrix AA is the set N(A)N(A) of all solutions of the homogeneous equation Ax=0Ax=0

The null space of a matrix AA is a subspace of Rn\Bbb{R}^n. The first question shows the proof of this.

To see if a vector uu is in N(A)N(A) (nullspace of AA), we simply compute:

Au=0 Au=0

If the product of AA and uu gives the zero vector, then it is in the null space of AA.

To find a basis for the null space of A, we have to:
1) Solve for Ax=0Ax=0.
2) Write the general solution in parametric vector form.
3) The vectors you see is a basis for N(A)N(A).

Note that the vectors in the basis are linearly independent.
  • 1.
    Null Space Overview:
    a)
    Definition of the Null space
    N(A)=N(A) = null space
    • A set of all vectors which satisfy the solution Ax=0Ax=0
    Definition of the Null space

    b)
    A vector u in the null space
    • Multiply matrix AA and vector uu
    Au=0Au=0 means uu is in the null space

    c)
    Finding a basis for the null space
    • Solve for Ax=0Ax=0
    • Write the general solution in parametric vector form
    • The vectors you see in parametric vector form are the basis of N(A)N(A).


  • 2.
    Showing that the null space of AA is a subspace
    Show that N(A)N(A) (null space of AA) is a subspace by verifying the 3 properties:
    1) The zero vector is in N(A)N(A)
    2) For each uu and vv in the set N(A)N(A), the sum of u+vu+v is in N(A)N(A) (closed under addition)
    3) For each uu in the set N(A)N(A), the vector cucu is in N(A)N(A). (closed under scalar multiplication)

  • 3.
    Verifying a vector is in the null space
    Is the vector is this matrix in null space in the null space of the matrix
    null space of this matrix

  • 4.
    Is the vector Verifying a vector is in the null space in the null space of the matrix
    null space of this matrix

  • 5.
    Finding a basis for the null space
    Find a basis for the null space of AA if:
    Finding a basis for the null space

  • 6.
    Find a basis for the null space of AA if:
    Finding a basis for the null space