Spanning Sets

In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

1. S=\set{\colvector{1\\2\\-1},\,\colvector{3\\-4\\2},\,\colvector{4\\-2\\1}}
2. Use $S$ and $W$ from the previous question. Is the vector $\colvector{6\\5\\-1}$ in $W$? Give an explanation of the reason for your answer.
3. A= \begin{bmatrix} 1 & 3 & 1 & 9\\ 2 & 1 & -3 & 8\\ 1 & 1 & -1 & 5 \end{bmatrix}

In we saw the solution set of a homogeneous system described as all possible linear combinations of two particular vectors. This is a useful way to construct or describe infinite sets of vectors, so we encapsulate the idea in a definition.

The span is just a set of vectors, though in all but one situation it is an infinite set. (Just when is it not infinite?) So we start with a finite collection of vectors $S$ ($p$ of them to be precise), and use this finite set to describe an infinite set of vectors, $\spn{S}$. Confusing the finite set $S$ with the infinite set $\spn{S}$ is one of the most persistent problems in understanding introductory linear algebra. We will see this construction repeatedly, so let us work through some examples to get comfortable with it. The most obvious question about a set is if a particular item of the correct type is in the set, or not in the set.

Having analyzed in , we will of course subject to a similar investigation.

With the notion of a span, we can expand our techniques for checking the consistency of a linear system. tells us a system is consistent if and only if the vector of constants is a linear combination of the columns of the coefficient matrix. This is because says that any solution to the system will provide a linear combination of the columns of the coefficient that equals the vector of constants. So consistency of a system is equivalent to the membership of the vector of constants in the span of the columns of the coefficient matrix. Read that last sentence again carefully. We will see this idea again, but more formally, in .

We will reprise , which is based on . We again make use of the matrix method .columns() to get all of the columns into a list at once.

You could try to find an example of a vector of constants which would create an inconsistent system with this coefficient matrix. But there is no such thing. Here is why the null space of coeff is trivial, just the zero vector.

The system is consistent, as we have shown, so we can apply . We can read as saying any two different solutions of the system will differ by an element of the null space, and the only possibility for this null space vector is just the zero vector. In other words, any two solutions cannot be different.

Condensed: the full section is in Beezer, A First Course in Linear Algebra.

We saw in that when a system of equations is homogeneous the solution set can be expressed in the form described by where the vector $\vect{c}$ is the zero vector. We can essentially ignore this vector, so that the remainder of the typical expression for a solution looks like an arbitrary linear combination, where the scalars are the free variables and the vectors are $\vectorlist{u}{n-r}$. Which sounds a lot like a span. This is the substance of the next theorem.

Here is an example that will simultaneously exercise the span construction and , while also pointing the way to the next section.

Condensed: the full section is in Beezer, A First Course in Linear Algebra.