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Linear Algebra

Rank, Nullity, and Dimension

5/12 in Linear Algebra. See all.

Row Rank
The row rank of a matrix is the dimension of the row space of the matrix. The row rank of a matrix is the maximum number of linearly independent rows in the matrix.
Unsurprisingly, the row rank is denoted .

Column Rank
The column rank of a matrix is the dimension of the column space of the matrix. The column rank of a matrix is the maximum number of linearly independent columns in the matrix.
If you had to guess how the column rank is denoted as, what would you say? ? You're right!

(and if you didn't say ... I don't know what to tell you.)

Rank
The rank of a matrix is, interestingly, the same as the row rank and the column rank.
There are a number of properties of the rank of a matrix:

  • The system is consistent if and only if . This is clear because it means that the columns of with are linearly dependent, so can be expressed as a linear combination of 's columns.
  • An matrix is invertible if and only if ( can be row-reduced to the identity matrix).
  • If we have an consistent -variable SLE , then the solutions to the system are expressible in unknowns (the number of zero rows after row-reducing).
  • If , then the rows of are linearly independent, and .
  • If is an matrix such that , then there exists matrix such that .
  • A square matrix is invertible if and only if it can be transformed to an upper triangular matrix with all diagonal elements equal to (no zero rows).
  • .

Column Space
Column space is the set of all possible outputs of . This is just the span of the columns (the basis vectors)! For any matrix , . This means that the rank is equal to the column rank and the row rank. We can find bases for the column space in one of three ways:

  1. By inspection.
  2. By transposing, row-reducing, and taking nonzero columns.
  3. By row-reducing (to ) and taking pivotal columns in 's corresponding columns in .

Row Space
The row space of a an matrix is the span of the rows of each viewed as a matrix in . The row rank of a matrix is the dimension of the row space. The two are denoted and . The basis of the row space of a matrix in row-reduced form is the nonzero rows of that matrix, and the row rank is the number of nonzero rows. Even better: if a matrix is obtained from matrix by applying some elementary row operations, then . Also, if has some LI columns, the corresponding columns in are LI.

Full Rank
Not a poker hand! Full rank is when the rank of a transformation is the same as the number of input dimensions (number of columns).

Finding a Basis for the...

  1. Row Space: Find the row-reduced form of the matrix. The basis is the set of nonzero rows.
  2. Image/Column Space: Find the row-reduced form of the transpose of the matrix. The basis is the set of nonzero rows.
  3. Kernel/Null Space: Find the row-reduced form of the matrix. The basis is the set of special solutions to the system . Here, row-reducing isn't actually necessary, but it's helpful to solve
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