Dimension and rank

Get the most by viewing this topic in your current grade. Pick your course now.

?
Intros
Lessons
  1. Dimension and Rank Overview:
  2. Dimension of a subspace
    • Dimension = number of vectors in the basis
    • Can we find dimension of column space and null space?
  3. Rank of a Matrix
    • Find the basis
    • Count the # of vectors
    • Shortcut = count the # of pivots
  4. Dimension of the Null Space
    • Find the general solution
    • Put in parametric vector form
    • Count the # of vectors
    • Shortcut = count the # of free variables
  5. The Rank Theorem
    • Rank A+A + dim N(A)=nN(A) = n
    • An example of using the theorem
?
Examples
Lessons
  1. Finding the Rank of a matrix
    Find the rank of AA if:
    Finding the Rank of a matrix
    1. 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
      1. Finding the dimension of the null space
        Find the dimension of the null space of AA if:
        Finding the dimension of the null space
        1. 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
          1. 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?
            1. 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?