LINEAR COMBINATIONS AND GENERATORS

Every vector space $V ≠$ {0} contains infinitely many vectors; for if $V$ contains a vector $𝒗$, it immediately must also contain all its multiples, i.e. $λ𝒗 ∈ V$ for each $λ ∈ ℝ$.

Consider the subspace $W =$ { $(x,ax)∣x ∈ ℝ$ } in $ℝ^2$
It is represented, in the Cartesian plane, by a line whose equation is $y = ax$. We can describe it, in an alternative way, as the set of all multiples of the vector $(1, a)$
$W =$ { $x(1,a) ∣ x∈ℝ$ }.

Graphically it is clear that if we know a point of a straight line (in the plane, but also in three-dimensional space) different from the origin, then we can immediately draw the line passing through it and the origin. We will see later that the fact of knowing the generators of a vector space allows us to determine it uniquely.

Now let us see another example. In $ℝ^2$, we consider the two vectors $(1,0)$ and $(0, 1)$.
We ask ourselves: what is the smallest subspace $W$ of $ℝ^2$ that contains both of these vectors? From the previous reasoning, we know that this subspace must contain the two subspaces $W_1$ and $W_2$ generated by $(1, 0)$ and $(0, 1)$:

• $W_1 =$ { $λ(1, 0)∣λ ∈ ℝ$ } represented by the $x$-axis
• $W_2 =$ { $μ(0, 1)∣μ ∈ ℝ$ } represented by the $y$-axis

Assume that $V$ is a vector space, $𝐯_1, ... , 𝐯_n$ are vectors of $V$ and $λ_1$, ..., $λ_n ∈ ℝ$. The vector $𝐰 = λ_1𝐯_1 +⋅⋅⋅+λ_n𝐯_n$ is said to be a linear combination of $𝐯_1,...,𝐯_n$ with scalars $λ_1,...,λ_n$.

We now come to the concept of vector space generated by some vectors, the main concept of this chapter along with that of linear independence.

Let $V$ be a vector space and let { $𝐯_1,...,𝐯_n$ } be a set of vectors of $V$.
The subspace generated (or spanned) by the vectors $𝐯_1, ... , 𝐯_n$ is the set of all their linear combinations, in symbols
$⟨𝐯_1,...,𝐯_n⟩ =$ { $λ_1𝐯_1 +⋅⋅⋅+ λ_n𝐯_n ∣ λ_1,...,λ_n ∈ℝ$ }.

If $V$ is a vector space and $𝐯 ∈ V$ , then the subspace generated by $𝐯$ is the set of multiples of $𝐯$, i.e. $⟨v⟩ =$ { $λ𝐯∣λ ∈ ℝ$ }.
Moreover, the subspace generated by the zero vector is the trivial subspace, which contains only the zero vector: $⟨𝟎⟩ =$ { $𝟎$ }.

Let $V$ be a vector space and let { $v_1, . . . , v_n$ } be a set of vectors of $V$. We say that $v_1,...,v_n$ generate $V$, or { $v_1,...,v_n$ } is a set of generators of $V$ if $V = ⟨v_1,...,v_n⟩$.

Example: The vectors $(1, 0)$ and $(0, 1)$ generate the vector space $ℝ^2$, as each vector $(a,b)$ of $ℝ^2$ can be written as a linear combination of $(1,0)$ and $(0, 1)$:

(a, b) = a(1, 0) + b(0, 1).

Assume that $V$ is a vector space, $𝐯_1, . . . , 𝐯_n$ are vectors of $V$ and $𝐰$ is a linear combination of them, namely: $𝐰 = λ_1𝐯_1 + ⋅ ⋅ ⋅ + λ_n𝐯_n. Then

⟨𝐯_1,...,𝐯_n⟩ = ⟨𝐯_1,...,𝐯_n,𝐰⟩

Conversely, if $⟨𝐯_1,...,𝐯_n⟩ = ⟨𝐯_1,...,𝐯_n,𝐰⟩$ then $𝐰$ is a linear combination of $𝐯_1, . . . , 𝐯_n$

Now we ask: How do we establish what the “redundant vectors” are in describing the subspace generated by a set of vectors? If we want to be efficient in describing a subspace, we must be able to describe it as the subspace generated by the smallest possible number of vectors. The answer to this question comes from the concept of linear independence.

Shortly, the story is: if a set of generators of a subspace is a set of linearly independent vectors, then we are sure that it is the most efficient way to describe the subspace generated by those vectors, namely that we are using the smallest number of vectors.

Let $V$ be a vector space. The vectors $𝐯_1, . . . , 𝐯_n ∈ V$ are linearly independent if for every linear combination
$λ_1·𝐯_1 +⋅⋅⋅+ λ_n·𝐯_n = 𝟎$,
we necessarily have that $λ_1 = ⋅ ⋅ ⋅ = λ_n = 0$

In other words, the only linear combination of the vectors $𝒗_1, . . . , 𝒗_n$ giving the zero vector is the one with all zero scalars.

We will say also that the set of vectors { $𝐯_1, . . . , 𝐯_n$ } is linearly independent.

The vectors $𝐯_1, . . . , 𝐯_n$ are linearly dependent, if they are not independent.
In other words, the vectors of the set { $𝐯_1, . . . , 𝐯_n$ } are linearly dependent if there exist scalars $λ_1,...,λ_n$, not all zero, such that $λ_1·𝐯_1+⋅⋅⋅+λ_n·𝐯_n = 𝟎$.

We note that, if a set of vectors contains the zero vector, then it is always a set of linearly dependent vectors.

A non-empty subset of a set of linearly independent vectors consists of vectors that are still linearly independent.