G.2 FOUR FUNDAMENTAL SUBSPACES
Consider again the case where matrix 
is real-valued and has rank r.
Definition: Column Space The column space of matrix A has dimension r and consists of all linear combinations of its columns. It is also called the range of A.
In the following expression, 
lies in the column space of A:
(G.11) 
where 
and {an} are the columns of A.
Definition: Null Space The null space of matrix A has dimension N−r and consists of all vectors x such that
(G.12) 
where 
. The null space is also known as the kernel of A, and its dimension is called the nullity.
Definition: Row Space The row space of matrix A has dimension r and consists of all linear combinations of the rows.
In the following expression, 
lies in the row space of A:
(G.13)
where and are the rows of A. The ...
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