Question: In Exercises, determine if the set of m vectors in three-dimensional space is linearly independent by solving for the scalars c1, c2 ... cm in Theorem II. Where appropriate, verify the result by using Theorem I.
Theorem II: Linear dependence and independence Let S be a set of non-zero n-vectors x1, x2,..., xm, with m ≥ 2. Then:
(a) Set S is linearly dependent if the vector equation
c1x1 + c2x2 + . . . + cmxm = 0
is true for some set of scalars (constants) c1, c2,..., cm that are not all zero;
(b) Set S is linearly independent if the vector equation
c1x1 + c2x2 + . . . + cmxm = 0
is only true when c1 = c2 =···= cm = 0.
Theorem I: Test for linear independence of vectors in three-dimensional space Let a, b, and c be any three vectors. Then the vectors are linearly independent if (a × b)· c 0, and they are linearly dependent if (a × b) · c = 0.
Exercise: 1. a = i - j + 3k, b = 2i - j + 2k, c = 3i + j + k, (m = 4)
2.a = i + j, b = j + k, c = i - k.