Reduced row-echelon form definition of Hoffman's book

Question

I'm reading Hoffman's linear algebra book and this is my first time studying linear algebra. I'm trying to understand this description of reduced row-echelon form. I see that $r$ represents rows of matrix but confused about $k_1,\ldots, k_r$ part.

Can some one explain what it means?

Does $k_1$ represents column of first row and if I put $k_1$ for $k_i$, does that mean all columns of that row or something else?

Thanks.

Answer 1

• $r$ is the number of non-zero rows.
• for each non-zero row $i$, $k_i$ is the column of the first non-zero entry.
• "$R_{ij} = 0$ for $i > r$" means that every row below row $r$, all entries are $0$. (There is an assumed "for all $j$" here, and similar assumptions for other statements.)
• "$R_{ij} = 0$ if $j < k_i$" means that on the non-zero rows, all entries to the left of $k_i$ must be $0$. (Note that the $i$ here is necessarily different than the $i$ in the first part of the statement, as that $i > r$, while $k_i$ is not defined unless $i \le r$.)
• "$R_{ik_j} = \delta_{ij}, 1\le i \le r, 1\le j\le r$" says the column of any $k_j$ will be $0$ except for $R_{jk_j} = 1$ (it only guarantees $0$ for rows up to $r$, but the first condition says rows below $r$ will also have $0$).
• "$k_1 < \dots < k_r$" says that the number of leading $0$s in the first $r$ rows (i.e., before the rows are entirely $0$) strictly increases with each row.

Answer 2

In a nutshell, it means,

$1.$ zero rows are at the bottom

$2.$ call first non-zero entry(from left) of a non-zero row, a 'leading entry', then $k_i$ basically represents columns with leading entries and $k_1<k_2<\cdots<k_r$

$3.$ a column with leading entry has all entries zero except leading entry