What is meant by a linearly independent set of vectors in #RR^n#? Explain?

Question

Hazel Anglin · Accepted Answer

A vector set #{a_1, a_2, ..., a_n}# is linearly independent, if there exists the set of scalars #{l_1, l_2,...,l_n}# for expressing any arbitrary vector #V# as the linear sum #sum l_i a_i, i=1,2,..n#.

Unit vectors in the directions of the frame of reference's axes are examples of linear independent sets of vectors. 2-D: #{i, j}#. Any arbitrary vector #a=a_1 i+a_2 j# 3-D: #{i, j, k}#. Any arbitrary vector #a=a_1 i+ a_2 j+a_3 k#.

Abigail Allen · Answer

A set of vectors #v_1,v_2,…,v_p# in a vector space #V# is said to be linearly independent #iff# the vector equation
#c_1v_1 + c_2v_2 + cdots+ c_pv_p = 0#
has only the trivial solution for #c_1 = c_2 = cdots =c_p = 0#.

Also, The set of vectors #{v_1, . . . , v_n} ⊂ V# is linearly independent #iff# (stands for iff) every vector #v ∈ "span"{v_1, . . . , v_n}# can be written uniquely as a linear combination
#v = a_1v_1 + · · · + a_nv_n#

Hope that helps...

Jace Anderson · Answer

undefined A set of vectors in $ \mathbb{R}^n $ is linearly independent if no vector in the set can be represented as a linear combination of the other vectors in the set. In other words, if for any vectors $ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k $ in the set and scalars $ c_1, c_2, \ldots, c_k $, the equation $ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \ldots + c_k\mathbf{v}_k = \mathbf{0} $ implies that $ c_1 = c_2 = \ldots = c_k = 0 $.