section-subspaces.ipynb

Kernel:

%%html
<link href="http://mathbook.pugetsound.edu/beta/mathbook-content.css" rel="stylesheet" type="text/css" />
<link href="https://aimath.org/mathbook/mathbook-add-on.css" rel="stylesheet" type="text/css" />
<style>.subtitle {font-size:medium; display:block}</style>
<link href="https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic" rel="stylesheet" type="text/css" />
<link href="https://fonts.googleapis.com/css?family=Inconsolata:400,700&subset=latin,latin-ext" rel="stylesheet" type="text/css" /><!-- Hide this cell. -->
<script>
var cell = $(".container .cell").eq(0), ia = cell.find(".input_area")
if (cell.find(".toggle-button").length == 0) {
ia.after(
    $('<button class="toggle-button">Toggle hidden code</button>').click(
        function (){ ia.toggle() }
        )
    )
ia.hide()
}
</script>

Important: to view this notebook properly you will need to execute the cell above, which assumes you have an Internet connection. It should already be selected, or place your cursor anywhere above to select. Then press the "Run" button in the menu bar above (the right-pointing arrowhead), or press Shift-Enter on your keyboard.

ParseError: KaTeX parse error: \newcommand{\lt} attempting to redefine \lt; use \renewcommand

Section20.2Subspaces

¶

Just as groups have subgroups and rings have subrings, vector spaces also have substructures. Let $V$ be a vector space over a field $F\text{,}$ and $W$ a subset of $V\text{.}$ Then $W$ is a of $V$ if it is closed under vector addition and scalar multiplication; that is, if $u, v \in W$ and $\alpha \in F\text{,}$ it will always be the case that $u + v$ and $\alpha v$ are also in $W\text{.}$

Example20.6

Let $W$ be the subspace of ${\mathbb R}^3$ defined by $W = \{ (x_1, 2 x_1 + x_2, x_1 - x_2) : x_1, x_2 \in {\mathbb R} \}\text{.}$ We claim that $W$ is a subspace of ${\mathbb R}^3\text{.}$ Since

$$\begin{align*} \alpha (x_1, 2 x_1 + x_2, x_1 - x_2) & = (\alpha x_1, \alpha(2 x_1 + x_2), \alpha( x_1 - x_2))\\ & = (\alpha x_1, 2(\alpha x_1) + \alpha x_2, \alpha x_1 -\alpha x_2), \end{align*}$$

$W$ is closed under scalar multiplication. To show that $W$ is closed under vector addition, let $u = (x_1, 2 x_1 + x_2, x_1 - x_2)$ and $v = (y_1, 2 y_1 + y_2, y_1 - y_2)$ be vectors in $W\text{.}$ Then

$$\begin{equation*} u + v = (x_1 + y_1, 2( x_1 + y_1) +( x_2 + y_2), (x_1 + y_1) - (x_2+ y_2)). \end{equation*}$$

Example20.7

Let $W$ be the subset of polynomials of $F[x]$ with no odd-power terms. If $p(x)$ and $q(x)$ have no odd-power terms, then neither will $p(x) + q(x)\text{.}$ Also, $\alpha p(x) \in W$ for $\alpha \in F$ and $p(x) \in W\text{.}$

Let $V$ be any vector space over a field $F$ and suppose that $v_1, v_2, \ldots, v_n$ are vectors in $V$ and $\alpha_1, \alpha_2, \ldots, \alpha_n$ are scalars in $F\text{.}$ Any vector $w$ in $V$ of the form

$$\begin{equation*} w = \sum_{i=1}^n \alpha_i v_i = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n \end{equation*}$$

is called a of the vectors $v_1, v_2, \ldots, v_n\text{.}$ The of vectors $v_1, v_2, \ldots, v_n$ is the set of vectors obtained from all possible linear combinations of $v_1, v_2, \ldots, v_n\text{.}$ If $W$ is the spanning set of $v_1, v_2, \ldots, v_n\text{,}$ then we say that $W$ is by $v_1, v_2, \ldots, v_n\text{.}$

Proposition20.8

Let $S= \{v_1, v_2, \ldots, v_n \}$ be vectors in a vector space $V\text{.}$ Then the span of $S$ is a subspace of $V\text{.}$

Proof

Let $u$ and $v$ be in $S\text{.}$ We can write both of these vectors as linear combinations of the $v_i$'s:

$$\begin{align*} u & = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n\\ v & = \beta_1 v_1 + \beta_2 v_2 + \cdots + \beta_n v_n. \end{align*}$$

Then

$$\begin{equation*} u + v =( \alpha_1 + \beta_1) v_1 + (\alpha_2+ \beta_2) v_2 + \cdots + (\alpha_n + \beta_n) v_n \end{equation*}$$

is a linear combination of the $v_i$'s. For $\alpha \in F\text{,}$

$$\begin{equation*} \alpha u = (\alpha \alpha_1) v_1 + ( \alpha \alpha_2) v_2 + \cdots + (\alpha \alpha_n ) v_n \end{equation*}$$

is in the span of $S\text{.}$