1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 347

EXAMPLE 2 Verifying Linear Independence of a Set of Vector

Functions

Show that t he set of vector functions

is linearly independent on the interval ( -oo, oo).

SOLUTION

Forming the matrix Y and computing det Y , we find

(

ex

detY = det 0

2 ex

= ex ( -e-x) (3) + e-x (1) (2ex) + 2(0) ( e-x)

• ex (1) ( e-x) - e-x (0) (3) - 2( -e-x) (2ex)

= - 3 + 2 + 0 - 1 - 0 + 4 = 2.

Since det Y I 0 for any x E (-oo, oo ), the given set of vector functions is
linearly independent on ( - oo, oo).

EXERCISES 8.1

For exercises 1-16, let

(

-1 2 1)
A= 0 3 -4 ' (

2 -1)

B = ~ ~ ' y=(-~),

and z = (1 -2).

State whether it is possible to compute the given expression or not.

When possible, compute the expression.

1. AB 2. BA 3. Ax 4. Ay 5. Az 6. Bx

7. By 8. Bz 9. xy 10. xz 11. yz^12. zy


8. A+ B 14. x + y 15. Ax+ y 16. By+ x