Review exercise 2.13. For the matrix

find its triangular factors and describe the conditions under which the columns of are linearly independent.

Answer: We start elimination by subtracting 1 times the first row from the second row, with

We next subtract 1 times the first row from the third row, with

and then subtract 1 times the first row from the fourth row, with

Turnng to the second column, we subtract 1 times the second row from the third row, with

and then subtract 1 times the second row from the fourth row, with

Finally we subtract 1 times the third row from the fourth row, with

We thus have

In order for the columns of and thus the columns of to be linearly independent, the values in the four pivot positions must be nonzero, so we must have , , , and .

Note that this does *not* mean that all four values must be nonzero, or that all fur values have to be unique. For example, the conditions would be satisfied if and so that

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.