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

2 Answers

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#.

Explanation:

Examples of linear independent set of vectors are unit vectors in the directions of the axes of the frame of reference, as given below.

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#.

Jun 5, 2016

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...