In exercises determine if the set of m vectors in

Question

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,..., x_m, with m ≥ 2. Then:

(a) Set S is linearly dependent if the vector equation

c₁x₁ + c₂x₂ + . . . + c_mx_m = 0

is true for some set of scalars (constants) c₁, c₂,..., c_m that are not all zero;

(b) Set S is linearly independent if the vector equation

c₁x₁ + c₂x₂ + . . . + c_mx_m = 0

is only true when c₁ = c₂ =···= c_m = 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.