When are vector v1 v2 vP in Rn linearly independent Provide

When are vector v_1, v_2 ... v_P in R^n linearly independent? (Provide the precise definition) How do you check if the columns of a matrix A are linearly independent? Be sure to understand why this in the case. Check if the column vectors (-4 0 1 2), (-3 -1 1 1), (0 1 -1 -2) are linearly independent. If not, provide an explicit non-trival linear combination. Without actually doing any calculation explain why (-4 0 1 2), (-3 -1 1 1), (0 1 -1 -2), (1 2 3 4), (5 6 7 8) must be linearly dependent. (Think about solution to Az = 0)

Solution

a) The vectors in a set {V₁, V₂ ... V_p} are said to be linearly independent if the equation c₁V₁ + c₂V₂ + ... + c_pV_p = 0, can only be sastified by c_i = 0 for i = 1,..., p.

This means that vectors in the set cannot be related or represented as a linear comnination with each others.

b) A set of vectors is linearly independent if the only representations of 0 as a linear combination of its vectors is the trivial representation in which all the scalars c_i are zero.

c) The set of vectors can be linear dependent as follows:

V1 + V2 + V3 = V4

where adding the three vectors give us a new solution, if there is a possibility of thinking in this way. But I don\'t see any specific linear combination by combining themself.

d) The 2 new vectors have a linear combination.