Question

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

Answer 1

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`.

Answer 2

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...