Problem

Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors i...

Use the method of Example 2.23 and Theorem 2.6 to determine if the sets of vectors in Exercise are linearly independent. If, for any of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a dependence relationship among the vectors.

Reference Example 2.23

Determine whether the following sets of vectors are linearly independent

In answering any question of this type, it is a good idea to see if you can determine by inspection whether one vector is a linear combination of the others. A little thought may save a lot of computation!

(a) The only way two vectors can be linearly dependent is if one is a multiple of the other. (Why?) These two vectors are clearly not multiples, so they are linearly independent.

(b) There is no obvious dependence relation here, so we try to find scalars c₁, c₂, c₃ such that

The corresponding linear system is

and the augmented matrix is

Once again, we make the fundamental observation that the columns of the coefficient matrix are just the vectors in question!

The reduced row echelon form is

(check this), so c₁=0, c₂=0, c₃=0. Thus, the given vectors are linearly independent. (c) A little reflection reveals that

so the three vectors are linearly dependent. [Set up a linear system as in part (b) to check this algebraically.]

(d) Once again, we observe no obvious dependence, so we proceed directly to reduce a homogeneous linear system whose augmented matrix has as its columns the given vectors:

If we let the scalars be c₁, c₂, and c₃, we have

from which we see that the system has infinitely many solutions. In particular, there must be a nonzero solution, so the given vectors are linearly dependent. If we continue, we can describe these solutions exactly: c₁ = –3c₃ and c₂ = 2c₃. Thus, for any nonzero value of c₃, we have the linear dependence relation