Dimension and rank

Dimension and rank

Lessons

Dimension of a Subspace
The dimension of a non-zero subspace SS (usually denoted as dim SS), is the # of vectors in any basis for SS. Since the null space and column space is a subspace, we can find their dimensions.

Note: Dimension of the column space = rank

Finding the Rank of a matrix:
1. Find the basis for the column space
2. Count the # of vectors in the basis. That is the rank.
Shortcut: Count the # of pivots in the matrix

Finding the dimensions of the null space:
1. Find the basis for the null space
2. Count the # of vectors in the basis. That is the dimension.
Shortcut: Count the # of free variables in the matrix.

The Rank Theorem
If a matrix AA has nn columns, then rank A+A+ dim N(A)=nN(A) = n.
  • 1.
    Dimension and Rank Overview:
    a)
    Dimension of a subspace
    • Dimension = number of vectors in the basis
    • Can we find dimension of column space and null space?

    b)
    Rank of a Matrix
    • Find the basis
    • Count the # of vectors
    • Shortcut = count the # of pivots

    c)
    Dimension of the Null Space
    • Find the general solution
    • Put in parametric vector form
    • Count the # of vectors
    • Shortcut = count the # of free variables

    d)
    The Rank Theorem
    • Rank A+A + dim N(A)=nN(A) = n
    • An example of using the theorem


  • 2.
    Finding the Rank of a matrix
    Find the rank of AA if:
    Finding the Rank of a matrix

  • 3.
    Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?
    What is the dimension of the subspace

  • 4.
    Finding the dimension of the null space
    Find the dimension of the null space of AA if:
    Finding the dimension of the null space

  • 5.
    Utilizing the Rank Theorem
    You are given the matrix AA and the echelon form of AA. Find the basis for the column space, and find the rank and the dimensions of the null space.
    Utilizing the Rank Theorem

  • 6.
    Understanding the Theorems
    Let the subspace of all solutions of Ax=0Ax=0 have a basis consisting of four vectors, where AA is 4×64 \times 6. What is the rank of AA?

  • 7.
    Let AA be a m×nm \times n matrix where the rank of AA is pp. Then what is the dimension of the null space of AA?