nLab fundamental theorem of linear algebra

see also rank-nullity theorem

Statement in operator form

For $f \colon V\to W$ a linear operator between vector spaces, where $V$ is finite-dimensional of dimension $n$, then the dimensions of the kernel and the image of $f$ are related by:

$$dim\,Ker\,f + dim\,Im\,f \;=\; n \,.$$

Statement in the matrix form

If $A$ is a finite rectangular matrix then the row rank (the dimension of the span of its rows (“row space”)) plus the dimension of the solution space of equation $A x = 0$ (“nullspace”) yields the number of columns. From the other side, column rank (the dimension of the space of columns) plus the dimension of the solution space of equation $y A = 0$ (or, equivalently, $A^T y^T = 0$) yields the number of rows.

Notice that now we have two statements as a rectangular matrix can be thought as a linear operator in two ways: acting by left multiplication on column vectors or acting by right multiplication on row vectors.

Terminology

Sometimes additional statements are included by some authors.

Most often the additional part quoted is that the row rank is the same as the column rank. This is a somewhat nontrivial statement in our formulation (dimension of the space of rows/columns); in some expositions of the theory the definitions of ranks are such that this statement is automatic. Namely, the rank is the minimal dimension $r$ such that the $m\times n$-matrix $A$ can be written in the form $V \Lambda^T$ where $\Lambda$ is $n\times r$, that is $\Lambda^T$ is $r\times n$, and $V$ is an $m\times r$ matrix. Then the transposed matrix can be written as $\Lambda V^T$. This is nothing than the image factorization of a linear map and its transpose which both go through the same intermediate space.

Literature

The following 2 refereces have somewhat nonstandard exposition of linear algebra to be amenable to constructive/algorithmic treatments useful in numerical linear algebra:

• Carl de Boer?, Linear algebra, pdf
• Carl de Boer?, Numerical functional analysis, University of Wisconsin lecture notes, chapter 1 (linear algebra), pdf

category: algebra