Page 211 - 35Linear Algebra

10.4 Review Problems 211

(b) If possible, write each vector as a linear combination of the pre-
ceding ones.
(c) Remove the vectors which can be expressed as linear combinations
of the preceding vectors to form a linearly independent ordered set.
(Every vector in your set set should be from the given set.)

4. Gaussian elimination is a useful tool to figure out whether a set of
vectors spans a vector space and if they are linearly independent.
Consider a matrix M made from an ordered set of column vectors
n
(v 1 , v 2 , . . . , v m ) ⊂ R and the three cases listed below:

(a) RREF(M) is the identity matrix.
(b) RREF(M) has a row of zeros.
(c) Neither case (a) or (b) apply.

First give an explicit example for each case, state whether the col-
umn vectors you use are linearly independent or spanning in each case.
Then, in general, determine whether (v 1 , v 2 , . . . , v m ) are linearly inde-
n
pendent and/or spanning R in each of the three cases. If they are
linearly dependent, does RREF(M) tell you which vectors could be
removed to yield an independent set of vectors?

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