Spanning sets

Lecture 12

Author

Affiliation

Minjae Park

Auburn University
MATH 2660 - Spring 2026

Published

February 6, 2026

Recap

A vector space \(V\) is a set equipped with vector addition and scalar multiplication, and is closed under linear combinations.
To check whether a set is a vector space, you need to verify that vector addition and scalar multiplication are well defined (this is usually straightforward), and that the set is closed under both operations (which together imply closure under linear combinations).
A subspace \(W\) of a vector space \(V\) is a subset of \(V\) that is itself a vector space.
To check whether a subset \(W\subset V\) is a subspace, it suffices to verify closure under linear combinations, since \(V\) already satisfies all other vector space properties.

Answer with T/F for each statement. For example, type TTTT if all statements are true.

The set of \(3\times3\) matrices \(V=M_{3\times3}\) is a vector space.
The set \(W=\{(x,y): x+y=1,\ x,y\in\mathbb{R}\}\) is a subspace of \(V=\mathbb{R}^2\).
The set of polynomials of degree exactly \(3\) is a vector space.
Let \(V\) be the set of differentiable functions and \(W\) be the set of continuous functions. Then \(W\) is a subspace of \(V\).

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Suppose we are given a set with well-defined vector addition and scalar multiplication.
If the set fails to be a vector space, it is often because some linear combinations of its elements are missing.
One way to construct a vector space is therefore to collect all linear combinations of a given set of vectors.
This idea leads naturally to the concept of a spanning set.

Let \(V\) be a vector space. A subset \(S=\{\vec{v}_1,\dots,\vec{v}_k\}\) of \(V\) is called a spanning set if every vector \(\vec{u}\in V\) can be written as a linear combination of vectors in \(S\). In this case, we say that \(S\) spans \(V\).
Given a set of vectors \(S\), the set of all linear combinations of vectors in \(S\) is called the span of \(S\), denoted by \[\mathop{\mathrm{span}}(S)=\{c_1\vec{v}_1+\cdots+c_k\vec{v}_k:\ c_i\in\mathbb{R}\}.\]
Equivalently, \(S\) is a spanning set of \(V\) if and only if \(\mathop{\mathrm{span}}(S)=V\).

Let \(V\) be the zero vector space \(\{\vec{0}\}\) (where “zero” may mean different things depending on the arithmetic, such as \(0\), \(\langle 0,0 \rangle\), or the zero matrix).
As a special convention, we define the empty set \(\emptyset=\{\}\) to be a spanning set of \(V\).
We interpret \(\vec{0}\) as an empty linear combination, so that \[\mathop{\mathrm{span}}(\emptyset)=\{\vec{0}\}.\]
This convention makes linear algebra more consistent, just as defining \(x^0=1\) makes exponent rules more consistent.
More formally, the span of a set \(S\) can be defined as the intersection of all subspaces that contain \(S\), which naturally includes this special convention.

Let \[W=\{\langle x,y,0 \rangle: x,y\in\mathbb{R}\},\] which is a vector space (in fact, a subspace of \(V=\mathbb{R}^3\)).
Since \[\mathop{\mathrm{span}}(\{\vec{e}_1,\vec{e}_2\})=\{x\vec{e}_1+y\vec{e}_2: x,y\in\mathbb{R}\} =\{\langle x,y,0 \rangle: x,y\in\mathbb{R}\},\] the set \(S_1=\{\langle 1,0,0 \rangle,\langle 0,1,0 \rangle\}\) is a spanning set of \(W\).
Spanning sets are not unique. For example, \[S_2=\{\vec{e}_1,-\vec{e}_2\}\] also spans \(W\).

A more interesting example is \[S_3=\{\langle 1,0,0 \rangle,\langle 1,1,0 \rangle\},\] since \[(x-y)\langle 1,0,0 \rangle+y\langle 1,1,0 \rangle=\langle x,y,0 \rangle.\]
We may also include redundant vectors, such as \[S_4=\{\langle 1,2,0 \rangle,\langle 1,0,0 \rangle,\langle 1,1,0 \rangle,\langle 0,1,0 \rangle\}.\] However, if we add \(\langle 0,0,1 \rangle\) (which is not in \(W\)) to any of above spanning sets, the span is no longer \(W\) but becomes \(V=\mathbb{R}^3\).

In the previous example, how can we check that \[\mathop{\mathrm{span}}(\{\langle 1,0,0 \rangle,\langle 1,1,0 \rangle\})=W = \{\langle x,y,0 \rangle: x,y\in\mathbb{R}\}?\]
First, verify that each vector in the set lies in \(W\).
Next, show that an arbitrary element of \(W\), say \(\langle x,y,0 \rangle\), can be written as a linear combination of these vectors.
This leads to the equation \[c_1\langle 1,0,0 \rangle+c_2\langle 1,1,0 \rangle=\langle x,y,0 \rangle.\]
We can solve this as a system of linear equations. If a solution exists (possibly infinitely many), then every element of \(W\) is a linear combination of the vectors in the set, and the set spans \(W\).

Find any spanning set of the vector space of \(\{\langle 0,y,z,0 \rangle: y,z \in\mathbb{R}\}\).
Find any spanning set of the vector space of \(3\times 3\) diagonal matrices.
Find any spanning set of the vector space of \(3\times 3\) upper triangular matrices.
Find any spanning set of the vector space of polynomials of degree at most \(2\) in variable \(x\).