Subspaces of Rn and Linear Independence

Subspaces

For any vector space, a subspace is a subset that is itself a vector space, under the inherited operations.

Important Lemma on Subspaces

For a nonempty subset $S$ of a vector space, under the inherited operations, the following are equivalent statements.

1. $S$ is a subspace of that vector space
2. $S$ is closed under linear combinations of pairs of vectors: for any vectors ${\vec {s}}_{1},{\vec {s}}_{2}\in S$ and scalars $r_{1},r_{2}$ the vector $r_{1}{\vec {s}}_{1}+r_{2}{\vec {s}}_{2}$ is in $S$
3. $S$ is closed under linear combinations of any number of vectors: for any vectors ${\vec {s}}_{1},\ldots ,{\vec {s}}_{n}\in S$ and scalars $r_{1},\ldots ,r_{n}$ the vector $r_{1}{\vec {s}}_{1}+\cdots +r_{n}{\vec {s}}_{n}$ is in $S$.

Briefly, the way that a subset gets to be a subspace is by being closed under linear combinations.

Proof:

"The following are equivalent" means that each pair of statements are equivalent.

$(1)\!\iff \!(2)\qquad (2)\!\iff \!(3)\qquad (3)\!\iff \!(1)$

We will show this equivalence by establishing that $(1)\implies (3)\implies (2)\implies (1)$. This strategy is suggested by noticing that $(1)\implies (3)$ and $(3)\implies (2)$ are easy and so we need only argue the single implication $(2)\implies (1)$.

For that argument, assume that $S$ is a nonempty subset of a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is closed under combinations of pairs of vectors. We will show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a vector space by checking the conditions.

The first item in the vector space definition has five conditions. First, for closure under addition, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_1,\vec{s}_2\in S } then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_1+\vec{s}_2\in S } , as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_1+\vec{s}_2=1\cdot\vec{s}_1+1\cdot\vec{s}_2 } .

Second, for any ${\vec {s}}_{1},{\vec {s}}_{2}\in S$, because addition is inherited from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } , the sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_1+\vec{s}_2 } in $S$ equals the sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_1+\vec{s}_2 } in $V$, and that equals the sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_2+\vec{s}_1 } in $V$ (because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a vector space, its addition is commutative), and that in turn equals the sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_2+\vec{s}_1 } in $S$. The argument for the third condition is similar to that for the second.

For the fourth, consider the zero vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } and note that closure of $S$ under linear combinations of pairs of vectors gives that (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s} } is any member of the nonempty set $S$) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\cdot\vec{s}+0\cdot\vec{s}=\vec{0} } is in $S$; showing that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{0} } acts under the inherited operations as the additive identity of $S$ is easy.

The fifth condition is satisfied because for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}\in S } , closure under linear combinations shows that the vector $0\cdot {\vec {0}}+(-1)\cdot {\vec {s}}$ is in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } ; showing that it is the additive inverse of ${\vec {s}}$ under the inherited operations is routine.

We usually show that a subset is a subspace with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2)\implies (1) } .

Example 1

The plane Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=\{\begin{pmatrix} x \\ y \\ z \end{pmatrix}\,\big|\, x+y+z=0\} } is a subspace of $\mathbb {R} ^{3}$. As specified in the definition, the operations are the ones inherited from the larger space, that is, vectors add in $P$ as they add in $\mathbb {R} ^{3}$

${\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}}+{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}}={\begin{pmatrix}x_{1}+x_{2}\\y_{1}+y_{2}\\z_{1}+z_{2}\end{pmatrix}}$

and scalar multiplication is also the same as it is in $\mathbb {R} ^{3}$. To show that $P$ is a subspace, we need only note that it is a subset and then verify that it is a space. Checking that $P$ satisfies the conditions in the definition of a vector space is routine. For instance, for closure under addition, just note that if the summands satisfy that $x_{1}+y_{1}+z_{1}=0$ and $x_{2}+y_{2}+z_{2}=0$ then the sum satisfies that $(x_{1}+x_{2})+(y_{1}+y_{2})+(z_{1}+z_{2})=(x_{1}+y_{1}+z_{1})+(x_{2}+y_{2}+z_{2})=0$.

Example 2

The $x$-axis in $\mathbb {R} ^{2}$ is a subspace where the addition and scalar multiplication operations are the inherited ones.

${\begin{pmatrix}x_{1}\\0\end{pmatrix}}+{\begin{pmatrix}x_{2}\\0\end{pmatrix}}={\begin{pmatrix}x_{1}+x_{2}\\0\end{pmatrix}}\qquad r\cdot {\begin{pmatrix}x\\0\end{pmatrix}}={\begin{pmatrix}rx\\0\end{pmatrix}}$

As above, to verify that this is a subspace, we simply note that it is a subset and then check that it satisfies the conditions in definition of a vector space. For instance, the two closure conditions are satisfied: (1) adding two vectors with a second component of zero results in a vector with a second component of zero, and (2) multiplying a scalar times a vector with a second component of zero results in a vector with a second component of zero.

Example 3

Another subspace of $\mathbb {R} ^{2}$ is

$\{{\begin{pmatrix}0\\0\end{pmatrix}}\}$

which is its trivial subspace.

Any vector space has a trivial subspace $\{{\vec {0}}\,\}$.

At the opposite extreme, any vector space has itself for a subspace. Template:AnchorThese two are the improper subspaces. Template:AnchorOther subspaces are proper.

Example 4

The condition in the definition requiring that the addition and scalar multiplication operations must be the ones inherited from the larger space is important. Consider the subset $\{1\}$ of the vector space $\mathbb {R} ^{1}$. Under the operations $1+1=1$ and $r\cdot 1=1$ that set is a vector space, specifically, a trivial space. But it is not a subspace of $\mathbb {R} ^{1}$ because those aren't the inherited operations, since of course $\mathbb {R} ^{1}$ has $1+1=2$.

Example 5

All kinds of vector spaces, not just $\mathbb {R} ^{n}$'s, have subspaces. The vector space of cubic polynomials $\{a+bx+cx^{2}+dx^{3}\,{\big |}\,a,b,c,d\in \mathbb {R} \}$ has a subspace comprised of all linear polynomials $\{m+nx\,{\big |}\,m,n\in \mathbb {R} \}$.

Example 6

This is a subspace of the $2\!\times \!2$ matrices

$L=\{{\begin{pmatrix}a&0\\b&c\end{pmatrix}}\,{\big |}\,a+b+c=0\}$

(checking that it is nonempty and closed under linear combinations is easy).

To parametrize, express the condition as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=-b-c} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L =\{\begin{pmatrix} -b-c &0 \\ b &c \end{pmatrix} \,\big|\, b,c\in\mathbb{R}\} =\{b\begin{pmatrix} -1 &0 \\ 1 &0 \end{pmatrix} +c\begin{pmatrix} -1 &0 \\ 0 &1 \end{pmatrix} \,\big|\, b,c\in\mathbb{R}\} }

As above, we've described the subspace as a collection of unrestricted linear combinations (by coincidence, also of two elements).

Span

The span(or linear closure) of a nonempty subset $S$ of a vector space is the set of all linear combinations of vectors from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [S] =\{ c_1\vec{s}_1+\cdots+c_n\vec{s}_n \,\big|\, c_1,\ldots, c_n\in\mathbb{R} \text{ and } \vec{s}_1,\ldots,\vec{s}_n\in S \} }

The span of the empty subset of a vector space is the trivial subspace. No notation for the span is completely standard. The square brackets used here are common, but so are "${\mbox{span}}(S)$" and "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{sp}(S)} ".

Lemma

In a vector space, the span of any subset is a subspace.

Proof:

Call the subset $S$. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } is empty then by definition its span is the trivial subspace. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is not empty, then we need only check that the span $[S]$ is closed under linear combinations. For a pair of vectors from that span, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}=c_1\vec{s}_1+\cdots+c_n\vec{s}_n } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w}=c_{n+1}\vec{s}_{n+1}+\cdots+c_m\vec{s}_m } , a linear combination

$p\cdot (c_{1}{\vec {s}}_{1}+\cdots +c_{n}{\vec {s}}_{n})+r\cdot (c_{n+1}{\vec {s}}_{n+1}+\cdots +c_{m}{\vec {s}}_{m})$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = pc_1\vec{s}_1+\cdots+pc_n\vec{s}_n +rc_{n+1}\vec{s}_{n+1}+\cdots+rc_m\vec{s}_m }

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } , $r$ scalars) is a linear combination of elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } and so is in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [S] } (possibly some of the ${\vec {s}}_{i}$'s forming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} equal some of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{s}_j} 's from ${\vec {w}}$, but it does not matter).

Example 1

The span of this set is all of $\mathbb {R} ^{2}$.

$\{{\begin{pmatrix}1\\1\end{pmatrix}},{\begin{pmatrix}1\\-1\end{pmatrix}}\}$

To check this we must show that any member of $\mathbb {R} ^{2}$ is a linear combination of these two vectors. So we ask: for which vectors (with real components $x$ and $y$) are there scalars Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2} such that this holds?

$c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}+c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}={\begin{pmatrix}x\\y\end{pmatrix}}$

Gauss' method

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \begin{array}{*{2}{rc}r} c_1 &+ &c_2 &= &x \\ c_1 &- &c_2 &= &y \end{array} &\xrightarrow[]{-\rho_1+\rho_2} &\begin{array}{*{2}{rc}r} c_1 &+ &c_2 &= &x \\ & &-2c_2 &= &-x+y \end{array} \end{array} }

with back substitution gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=(x-y)/2} and $c_{1}=(x+y)/2$. These two equations show that for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} that we start with, there are appropriate coefficients $c_{1}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2} making the above vector equation true. For instance, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1} and $y=2$ the coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=-1/2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1=3/2} will do. That is, any vector in $\mathbb {R} ^{2}$ can be written as a linear combination of the two given vectors.