I'm currently studying triangular forms and factorizations, and have gotten in a spot of trouble with the notation my text is using (Matrix Analysis), particularly in reference to the row/column inclusion property relating to LU factorization.
It reads: "$A\in M_n$ has an LU factorization in which L is nonsingular if and only if A has the row inclusion property: For each $i=1,...,n-1, A[\{i+1,1,...,i\}]$ is a linear combination of the rows of $A[\{ 1,...,i\}]$.
What is the $A[\{1,...,i\}]$ notation expressing? The text later talks about the determinant of this, so I wouldn't think that it's just the first through $i$ rows of $A$.
It appears that this is notation meant to convey specific submatrices through indexing rows and columns. For example, if we let $A=\left[\begin{matrix}1&2&3\\4&5&6\\7&8&9\\ \end{matrix}\right]$, $A[\{1,3\},\{1,2,3\}]=\left[\begin{matrix}1&2&3\\7&8&9\\ \end{matrix}\right]$ as we took the first and third row and the first, second, and third column of $A$.
Hence, this theorem says that the submatrix of $A$ formed by only keeping the $i+1$ row and the $\{1,...,i\}$ columns is a linear combination of the rows of the leading submatrix $A[\{1,...,i\}]$. This can also be stated that for any leading submatrix of $A$, the row of equal size directly below it is linearly dependent on the rows of the given submatrix.