In our previous lecture, we said that subspaces of $\large \mathbb{R}^n$ play a large role in linear algebra. Today, we'll introduce one of the most important subspaces.

Recall, a weighted sum of vectors $\large c_1\vec{v_1}+c_2\vec{v_2}+\dots+c_m\vec{v_m}$ is called a linear combination. It's natural to wonder what points we can reach through a linear combination of given vectors, a question that leads to the big definition of this section.

Here's our word of the day:

Definition: The span of $\large \vec{v_1}, \vec{v_2},\dots, \vec{v_n}$ is the set of all linear combinations of $\large \vec{v_1}, \vec{v_2},\dots,\vec{v_n}$. We write $\large\text{Span}\lbrace \vec{v_1}, \vec{v_2},\dots, \vec{v_n}\rbrace$.

Let's look at a few examples.

First, let $\large \vec{v_1}=\left< 2 , 5 \right>$. What is $\large \text{Span}\lbrace \vec{v_1}\rbrace$, the span of a single vector? This is the collection of all the vectors $\large c\vec{v_1}$ for all the possible values of $\large c$. This is just all the possible scalings of the original vector, which is just a line. See the image:

Next, what is $\large \text{Span}\lbrace\vec{v_1},\vec{v_2}\rbrace$ if $\large \vec{v_1}=\left< 2,-1,1\right>$ and $\large \vec{v_2}=\left< 1,0,3\right>$? The images below show, from left to right, (1) the two vectors, (2) the two vectors and a linear combination of them, (3) the two vectors, five linear combinations of them, and the plane containing the two vectors.

We see that the span of the two vectors is the plane containing them.

One question of interest is whether a certain vector is included in the span of a given set of vectors. This question comes down to solving a linear system of equations, as we'll see in the following examples.

• Can $\large \vec{u} = \left< 7, 4, -3\right>$ be written as a linear combination of $\large \vec{v_1}=\left< 1, -2, -5\right>$ and $\large \vec{v_2}=\left<2, 5, 6\right>$? In other words, is $\large \vec{u}\in \text{Span}\lbrace \vec{v_1},\vec{v_2} \rbrace$?

We see in the video that the augmented matrix we build to answer a question like this is (putting the vectors in as columns):

$$\large \begin{bmatrix} \vec{v_1} & \vec{v_2} & : & \vec{u} \\ \end{bmatrix} $$

• Is $\large \left<-3, 8, 1\right>$ in the span of $\large \left< 1, -2, 3\right>$ and $\large \left<5,-13,-3\right>$?

We'll finish with a key fact.

Theorem: If $\large \vec{v_1}, \vec{v_2}, \dots, \vec{v_m}$ are all vectors in $\large \mathbb{R}^n$, then $\large \text{Span}\lbrace \vec{v_1},\vec{v_2}\dots, \vec{v_m} \rbrace$ is a subspace of $\large \mathbb{R}^n$.

To see that this is true, we just need to look at three things. For brevity, let's use $\large H$ for the span.

• Is $\large \vec{0}\in H$? Yes, the span of a set of vectors includes all possible linear combinations $\large c_1\vec{v_1}+c_2\vec{v_2}+\dots+c_m\vec{v_m}$ of the vectors. If all the constants in the linear combination are zeros, the linear combination will be the zero vector. So, yes, $\large \vec{0}\in H$.
• Is $\large H$ closed under addition? In other words, if $\large \vec{u}\in H$ and $\large \vec{v} \in H$, is $\large \vec{u}+\vec{v}$ also in $\large H$? Yes, and we'll see this in the video below.
• Is $\large H$ closed under scalar multiplication? In other words, if $\large \vec{u}\in H$, is $\large c\vec{u}$ also in $\large H$ for any scalar $\large c$? Yes, and we'll see this in the video below.

# Section Outcomes

I can:

• Recall and use the definition of the span of a set of vectors.
• Determine whether a given vector is part of the span of a set of vectors.