Linear Algebra 6: Rank, Basis, Dimension

This is a continuation of my Linear Algebra series, which should be viewed as an extra resource while going along with Gilbert Strang’s class 18.06 on OCW. This can be closely matched to Lecture 9 and 10 in his series.


To put it simply, the rank of the matrix represents the amount of independent columns in the matrix. This number, r, is very important when examining a matrix. Let’s take the rank of this matrix.

  • n - r is the amount of dependent / free columns, as well as free variables
  • r is the amount of vectors that define the column space


The basis is the smallest set of vectors possible that can be used to describe a vector space. A vector space has an infinite amount of bases. For example, all of the following are basis vectors of R².

  • The vectors must span the space in question.


Dimension is possibly the simplest concept — it is the amount of dimensions that the columns, or vectors, span. The dimension of the above matrix is 2, since the column space of the matrix is 2.

Full Rank; r = m = n

Often we deal with the case of full rank: where r = m = n. In this case, our matrix is obviously square, since it requires that m = n. Furthermore, this means that every column (and every row) is independent.

Full Column Rank; r = n, r < m

In full column rank, your matrix is made up of fully independent columns (since r = n, or one pivot in each column). The difference is that you have dependent rows, or leftover rows.

Pivots are highlighted. Forgot to have a “2” at the end of the last b in c.

Full Row Rank; r = m, r < n

Full row rank is when our equation has the same amount of pivots as rows. In this scenario, our matrix does have free variables and free columns, and thus has entries in the null space.


In full rank matrices, or r = m = n

  • There is one unique solution to every b.
  • The reduced-row echelon form R is the identity I.
  • There is nothing in the null space
  • The matrix is invertible
  • The reduced-row echelon form R is the identity I on top of a zero matrix
  • There is nothing in the null space
  • The reduced-row echelon form R is the identity I to the left of a zero matrix.
  • There is n-r special solutions in the null space.

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