Linear Independence of a Vector Space

Question

Is the zero vector space v={0}, linearly independent. If so, why is that?

Answer 1

Linear dependence means that for a sequence of vectors, {v1,v2,...vn} from the vector space, there exists a sequence of scalars {a1,a2,...an} not all zero such that

a₁v₁ + a₂v₂ + ... + a_nv_n = 0. (1)

For linear independence, a₁,a₂,...,a_n must all be zero. Or in other words, the equation (1) must only have the trivial solution. No vector can be written as a linear combination of the other vectors.

In this case, you are looking at the zero vector in R¹ or one dimensional space. You have a set of vectors, with just one vector in it, such that for any scalar a₁, with that scalar not equal zero a₁v₁ = 0. This will work since the vector itself is zero.

So by definition, the null space, or zero vector space is linearly dependent.

Note that if you are looking at another set, which has other vectors and also the zero vector, it will not be linearly independent. This is because you can write

a₁v₀ + 0v₁ +0v₂ +...+ 0v_n = 0, with a₁ not equal zero.