Linear Algebra: Linear Combinations

Span


The span of a set of vectors $v_1, \ldots, v_n$ in a vector space $V$ over $F$ is the set of all possible linear combinations of $v_1, \ldots, v_n$. The span of a set of vectors does not have a fancy symbol to denote it, and is instead written as $\text{span}(v_1, \ldots, v_n)$. What a plain bagel! In set builder notation, we can write the span out like this:

$$\text{span}(v_1, \ldots, v_n) = \left\{\sum\limits_{i}^{n} a_i v_i : a_i \in F \right\}$$

The span of an empty set of vectors is defined to be $\{ 0 \}$, the set containing only the zero vector.

If the span of a set of vectors $v_1, \ldots, v_n$ is equal to $V$, then the vectors are said to span $V$.

A vector space $V$ is called a finite-dimensional vector space if it is spanned by a finite number of vectors. Otherwise it is called an infinite-dimensional vector space.


Problems

  1. Determine whether $\langle 3, 2 \rangle \in \text{span}\left( \langle 1, 1 \rangle, \langle 1, 2 \rangle \right)$.

    Determining whether a vector is an element of the span of some vectors is the same as determining whether it's a linear combination of those vectors. So if you read the previous section on linear combinations and though, "Aw gee, that looks easy, I'll skip it," well guess what, double whammy!

    $ \langle 3, 2 \rangle = a_1 \langle 1, 1 \rangle + a_2 \langle 1, 2 \rangle \\ \langle 3, 2 \rangle = \langle a_1, a_1 \rangle + \langle a_2, 2a_2 \rangle \\ \langle 3, 2 \rangle = \langle a_1 + a_2, a_1 + 2a_2 \rangle \\ $

    There are two equations and two unknowns, so this vector has a shot of being in that span:

    $ 3 = a_1 + a_2 \\ a_1 = 3 - a_2 \\ $

    $ 2 = a_1 + 2a_2 \\ 2 = 3 - a_2 + 2a_2 \\ -1 = a_2 \\ $

    $ a_1 = 3 - (-1) \\ a_1 = 4 \\ $

    Looks like we've got a winner, folks. That vector is indeed part of the span:

    $ \langle 3, 2 \rangle = 4 \langle 1, 1 \rangle - \langle 1, 2 \rangle \\$

    It must feel good to belong to something important.

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  2. Determine whether $\langle 2, 3 \rangle \in \text{span}\left(\langle 1, i \rangle, \langle 0, i \rangle \right)$.

    $ \langle 2, 3 \rangle = a_1 \langle 1, i \rangle + a_2 \langle 0, i \rangle \\ \langle 2, 3 \rangle = \langle a_1, ia_1 \rangle + \langle 0, ia_2 \rangle \\ \langle 2, 3 \rangle = \langle a_1, ia_1 + ia_2 \rangle \\ $

    There are two equations and two unknowns, so the vector might be in the span. Let's solve and find out:

    $ 2 = a_1 $

    $ 3 = ia_1 + ia_2 \\ 3 = 2i + ia_2 \\ 3 - 2i = ia_2 \\ \dfrac{3}{i} - 2 = a_2 \\ $

    Well, it's kind of awkward and gangly, like most complex numbers, but it'll do:

    $ \langle 2, 3 \rangle = 2 \langle 1, i \rangle + \left(\dfrac{3}{i} -2\right) \langle 0, i \rangle \\$ $

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  3. Show that the span of a set of vectors is a subspace.

    To show that any set is a subspace, they must fulfill the three requirements outlined in the Subspaces section. Let $v_1, \ldots, v_n$ be vectors in a vector space $V$ over $F$.

    Additive identity: $0$ is a linear combination of $v_1, \ldots, v_n$, namely $0 = 0v_1 + \ldots 0v_n$. Therefore $0 \in \text{span}\left( v_1, \ldots, v_n \right)$.

    Closed under addition: Let $a$ and $b$ be in $\text{span}(v_1, \ldots, v_n)$ such that $a = a_1v_1 + \ldots + a_nv_n$ and $b = b_1v_1 + \ldots + b_nv_n$, where each $a_i$ and $b_i$ are in $F$. Then

    $ a + b = (a_1v_1 + \ldots + a_nv_n) + (b_1v_1 + \ldots + b_nv_n) \\ a + b = (a_1 + b_1)v_1 + \ldots + (a_n + b_n)v_n $

    Therefore $a + b$ is also in $\text{span}(v_1, \ldots, v_n)$, so $\text{span}(v_1, \ldots, v_n)$ is closed under addition.

    Closed under addition: Let $a$ be in $\text{span}(v_1, \ldots, v_n)$ such that $a = a_1v_1 + \ldots + a_nv_n$, and let $c \in F$. Then

    $ ca = c(a_1v_1 + \ldots + a_nv_n) \\ca = (ca_1)v_1 + \ldots + (ca_n)v_n$

    Therefore $ca$ is also in $\text{span}(v_1, \ldots, v_n)$, so $\text{span}(v_1, \ldots, v_n)$ is closed under scalar multiplication.

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  4. Show that the span of a set of vectors is the smallest subspace containing all the vectors in the set.

    Let $v_1, \ldots, v_n$ be a set of vectors in a vector space $V$ over a field $F$.

    Each vector $v_i$ is in $\text{span}(v_1, \ldots, v_n)$, as $v_i = 1v_i + \displaystyle\sum\limits_{j \neq i} 0 v_j$.

    Conversely, as subspaces are closed under scalar addition and multiplication, each subspace containing all of $v_1, \ldots, v_n$ contains all linear combinations of $v_1, \ldots, v_n$ and therefore contains $\text{span}(v_1, \ldots, v_n)$.

    Therefore, $\text{span}(v_1, \ldots, v_n)$ is the smallest subspace of $V$ containing all of $v_1, \ldots, v_n$.

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  5. Let $F^n$ be a vector space over $F$, where $F$ is a field (such as $\mathbb{R}$ or $\mathbb{C}$). Consider the vectors $e_1, \ldots, e_n$, where $e_1 = \langle 1, 0, \ldots, 0 \rangle, e_2 = \langle 0, 1, \ldots, 0 \rangle, \ldots, e_n = \langle 0, 0, \ldots, 1 \rangle$. Show that these vectors span $F^n$.

    By the definition of span, we see that

    $ \text{span}(e_1, e_2, \ldots, e_n) = \left\{ c_1e_1 + c_2e_2 + \ldots + c_ne_n : c_i \in F \right\} \\ $

    From here we just simplify and follow our noses:

    $ \text{span}(e_1, e_2, \ldots, e_n) = \left\{ \langle c_1, 0, \ldots, 0 \rangle + \langle 0, c_2, \ldots, 0 \rangle + \ldots + \langle 0, 0, \ldots, c_n \rangle : c_1, c_2, \ldots, c_n \in F \right\} \\ \text{span}(e_1, e_2, \ldots, e_n) = \left\{ \langle c_1, c_2, \ldots, c_n \rangle : c_1, c_2, \ldots, c_n \in F \right\} \\ \text{span}(e_1, e_2, \ldots, e_n) = F^n \\ $

     

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