Page 377 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

taken prisoner by the Russian army, and on his release studied philosophy at various German universities. He became a French
citizen in 1800 and eventually settled in Paris, where he did research in analysis leading to some controversial mathematical
papers and relatedly to a famous court trial over financial matters. Several years thereafter, his proposed research on the
determination of longitude at sea was rebuffed by the British Board of Longitude, and Wrónski turned to studies in Messianic
philosophy. In the 1830s he investigated the feasibility of caterpillar vehicles to compete with trains, with no luck, and spent
his last years in poverty. Much of his mathematical work was fraught with errors and imprecision, but it often contained
valuable isolated results and ideas. Some writers attribute this lifelong pattern of argumentation to psychopathic tendencies
and to an exaggeration of the importance of his own work.

THEOREM 5.3.4

If the functions have continuous derivatives on the interval , and if the Wronskian of these

functions is not identically zero on , then these functions form a linearly independent set of vectors in

.

EXAMPLE 9 Linearly Independent Set in

Show that the functions and form a linearly independent set of vectors in .

Solution

In Example 8 we showed that these vectors form a linearly independent set by noting that neither vector is a scalar multiple of the
other. However, for illustrative purposes, we shall obtain this same result using Theorem 5.3.4. The Wronskian is

This function does not have value zero for all x in the interval , as can be seen by evaluating it at , so and
form a linearly independent set.

EXAMPLE 10 Linearly Independent Set in

Show that , , and form a linearly independent set of vectors in .

Solution

The Wronskian is

This function does not have value zero for all x (in fact, for any x) in the interval , so , , and form a linearly