How do I explain (in words) that basis B spans V, is linearly independent, and is a basis of V?

Question

Let V be a vector space and basis B = {v1, . . . , vn} a subset of V . In complete sentences, explain what it means that

(a) B spans V , if for every v ∈ V there are coefficients a1, . . . , an ∈ R, so that v = a1v1 + · · · + anvn.

(b) B is linearly independent if a1v1 + · · · + anvn = 0 implies that a1 = · · · = an = 0.

(c) B is a basis of V if B is linearly independent and B spans V .

Answer 1

First, a bit of vocabulary:

A linear combination of the vectors v₁,...,v_n ∈ V is a₁v₁ + ... + a_nv_n, where a₁,...,a_n ∈ R

Note that a linear combinations of the vectors v₁,...,v_n ∈ V is itself a vector ∈ V

An equation of dependency of the vectors v₁,...,v_n ∈ V is an equation where the left-hand side is a linear combination of v₁,...,v_n ∈ V, and the right-hand side is the zero vector: a₁v₁ + ... + a_nv_n = 0

Clearly, this equation is true if a₁ = ... = a_n = 0; this is called the trivial solution.

The span of the vectors v₁,...v_n ∈ V is the set of all linear combinations of those vectors.

The notation: span{v₁,...,v_n} = {a₁v₁ + ... + a_nv_n : a₁,...,a_n ∈ R}

We say that {v₁,...,v_n} is a spanning set for V if V = span{v₁,...,v_n}

Putting this together, for B = {v₁,...,v_n}, we have:

(a) "B spans V" means "{v₁,...,v_n} is a spanning set for V"

(b) "B is linearly independent" means "The only solution to an equation of dependency for v₁,...,v_n is the trivial solution"

(c) "B is a basis for V" means "{v₁,...,v_n} is a linearly independent spanning set for V"