I have been watching 3blue1brown's Essence of Linear Algebra series on youtube, and I have a question about $2 \times 3$ matrices. For example: \begin{bmatrix}3&1&4\\1&5&9\end{bmatrix}

Although the basis vectors are linearly independent, the matrix transforms 3D space into 2D space. Thus, would this matrix be considered full rank or not?

  • 5
    $\begingroup$ What do you mean by "full rank" here? The matrix has rank $2$, which is the largest possible rank for a $2\times 3$ matrix. $\endgroup$ Nov 23, 2018 at 16:31
  • 4
    $\begingroup$ This is purely a matter of semantics: how have you defined "full rank"? $\endgroup$ Nov 23, 2018 at 16:37

1 Answer 1


Full row rank means that the rows are linearly independent and full column rank means that the columns are linearly independent.

For a square matrix we say the matrix is full rank if all rows and columns are linearly independent. For a non-square matrix, either the columns or the rows are linearly dependent (whichever is larger).

To say that a non-square matrix is full rank is to usually mean that the row rank and column rank are as high as possible.

In the example in the question there are three columns and two rows. the matrix is full rank if the matrix is full row rank.


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