[Math] Can rectangular matrix be a triangular matrix
linear algebra
I am confused with what is mentioned here:
Linear algebra and its applications-David C. Lay
Update: This question may be closed. My comment to the answer below clarifies my misunderstanding with the text.
Best Answer
I'm not entirely sure what you're asking, but to answer your title question: a triangular matrix is a special kind of square matrix, while a rectangular matrix is a matrix for which horizontal and vertical dimensions are not the same (http://mathworld.wolfram.com/RectangularMatrix.html)
I generally liked teaching out of an earlier edition of Poole (several years ago). I found I had time to do, and liked, some of his applications at the end of the sections. The students seemed to do well with the book. On the other hand (at least in the earlier editions--I haven't seen the more recent edition) the definition and discussion of general vector spaces was so late in the book that I had very little time for it. Also, the book was quite pricey at the time.
To summarize the comments into an answer:
The matrix
$$\begin{pmatrix}1&2&3\\0&4&5\end{pmatrix} $$
is echelon, but not triangular (because not square).
The matrix
$$\begin{pmatrix}1&2&3\\0&0&4\\0&0&5\end{pmatrix} $$
is triangular, but not echelon (because the leading entry $5$ is not to the right of the leading entry $4$).
However, for non-singular square matrices, "row echelon" and "upper triangular" are equivalent.
Best Answer
I'm not entirely sure what you're asking, but to answer your title question: a triangular matrix is a special kind of square matrix, while a rectangular matrix is a matrix for which horizontal and vertical dimensions are not the same (http://mathworld.wolfram.com/RectangularMatrix.html)