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.

Lemma 1

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 $V$ and that $S$ is closed under combinations of pairs of vectors. We will show that $S$ 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 ${\vec {s}}_{1},{\vec {s}}_{2}\in S$ then ${\vec {s}}_{1}+{\vec {s}}_{2}\in S$, as ${\vec {s}}_{1}+{\vec {s}}_{2}=1\cdot {\vec {s}}_{1}+1\cdot {\vec {s}}_{2}$.

Second, for any ${\vec {s}}_{1},{\vec {s}}_{2}\in S$, because addition is inherited from $V$, the sum ${\vec {s}}_{1}+{\vec {s}}_{2}$ in $S$ equals the sum ${\vec {s}}_{1}+{\vec {s}}_{2}$ in $V$, and that equals the sum ${\vec {s}}_{2}+{\vec {s}}_{1}$ in $V$ (because $V$ is a vector space, its addition is commutative), and that in turn equals the sum ${\vec {s}}_{2}+{\vec {s}}_{1}$ in $S$. The argument for the third condition is similar to that for the second.

For the fourth, consider the zero vector of $V$ and note that closure of $S$ under linear combinations of pairs of vectors gives that (where ${\vec {s}}$ is any member of the nonempty set $S$) $0\cdot {\vec {s}}+0\cdot {\vec {s}}={\vec {0}}$ is in $S$; showing that ${\vec {0}}$ acts under the inherited operations as the additive identity of $S$ is easy.

The fifth condition is satisfied because for any ${\vec {s}}\in S$, closure under linear combinations shows that the vector $0\cdot {\vec {0}}+(-1)\cdot {\vec {s}}$ is in $S$; 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 $(2)\implies (1)$.

Example 1

The plane $P=\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\,{\big |}\,x+y+z=0\}$ 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 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} 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 $a=-b-c$.

$L=\{{\begin{pmatrix}-b-c&0\\b&c\end{pmatrix}}\,{\big |}\,b,c\in \mathbb {R} \}=\{b{\begin{pmatrix}-1&0\\1&0\end{pmatrix}}+c{\begin{pmatrix}-1&0\\0&1\end{pmatrix}}\,{\big |}\,b,c\in \mathbb {R} \}$

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 $S$.

$[S]=\{c_{1}{\vec {s}}_{1}+\cdots +c_{n}{\vec {s}}_{n}\,{\big |}\,c_{1},\ldots ,c_{n}\in \mathbb {R} {\text{ and }}{\vec {s}}_{1},\ldots ,{\vec {s}}_{n}\in S\}$

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 "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{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 2

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

Proof:

Call the subset 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 } . 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 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 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

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\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 } , 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 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 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}_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 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}} , but it does not matter).

Example 7

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 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 \mathbb{R}^2} is a linear combination of these two vectors. So we ask: for which vectors (with real components 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} ) 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?

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\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 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=(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 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} 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 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=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 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 \mathbb{R}^2} can be written as a linear combination of the two given vectors.

Linear Independence

We first characterize when a vector can be removed from a set without changing the span of that set.

Lemma 3

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 S } is a 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} ,

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]=[S\cup\{\vec{v}\}] \quad\text{if and only if}\quad \vec{v}\in[S] }

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{v}\in V} .

Proof: The left to right implication is easy. 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]=[S\cup\{\vec{v}\}]} then, since 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}\in[S\cup\{\vec{v}\}] } , the equality of the two sets gives 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{v}\in[S] } .

For the right to left implication assume 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{v}\in [S] } to 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]=[S\cup\{\vec{v}\}] } by mutual inclusion. The inclusion 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 is obvious. For the other inclusion <math> [S]\supseteq[S\cup\{\vec{v}\}] } , write an element 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\cup\{\vec{v}\}] } 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 d_0\vec{v}+d_1\vec{s}_1+\dots+d_m\vec{s}_m } and substitute 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} } 's expansion as a linear combination of members of the same set 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 d_0(c_0\vec{t}_0+\dots+c_k\vec{t}_k)+d_1\vec{s}_1+\dots+d_m\vec{s}_m } . This is a linear combination of linear combinations and so distributing 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 d_0 } results in a linear combination 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 } . Hence each member 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\cup\{\vec{v}\}]} is also a member 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]} .

Example: 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 \mathbb{R}^3 } , 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{v}_1=\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\quad \vec{v}_2=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\quad \vec{v}_3=\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} }

the spans 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}_1,\vec{v}_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 [\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}] } are equal since 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}_3 } is in the 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}_1,\vec{v}_2\}] } .

Lemma 2 says that if we have a spanning set then we can remove a 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}} to get a new set 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} with the same span if and only 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{v}} is a linear combination 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} . Thus, under the second sense described above, a spanning set is minimal if and only if it contains no vectors that are linear combinations of the others in that set. We have a term for this important property.

Definition of Linear Independence

A subset of a vector space is linearly independent if none of its elements is a linear combination of the others. Otherwise it is linearly dependent. }}

Here is an important observation:

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}_0=c_1\vec{s}_1+c_2\vec{s}_2+\cdots +c_n\vec{s}_n }

although this way of writing one vector as a combination of the others visually sets 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}_0 } off from the other vectors, algebraically there is nothing special in that equation about 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}_0 } . For any ${\vec {s}}_{i}$ with a coefficient $c_{i}$ that is nonzero, we can rewrite the relationship to set off 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}_i } .

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}_i=(1/c_i)\vec{s}_0+(-c_1/c_i)\vec{s}_1 +\dots+(-c_n/c_i)\vec{s}_n }

When we don't want to single out any vector by writing it alone on one side of the equation, we will instead say 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{s}_0,\vec{s}_1,\dots,\vec{s}_n } are in a linear relationship and write the relationship with all of the vectors on the same side. The next result rephrases the linear independence definition in this style. It gives what is usually the easiest way to compute whether a finite set is dependent or independent.

Lemma 4

A subset 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 } of a vector space is linearly independent if and only if for any distinct 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,\dots,\vec{s}_n\in S } the only linear relationship among those vectors

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\vec{s}_1+\dots+c_n\vec{s}_n=\vec{0} \qquad c_1,\dots,c_n\in\mathbb{R} }

is the trivial one: 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=0,\dots,\,c_n=0 } . Proof: This is a direct consequence of the observation above.

If the set 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 linearly independent then no vector 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}_i} can be written as a linear combination of the other 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} so there is no linear relationship where 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}\,} 's have nonzero coefficients. 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 linearly independent then some 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}_i } is a linear combination 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}_i=c_1\vec{s}_1+\dots+c_{i-1}\vec{s}_{i-1} +c_{i+1}\vec{s}_{i+1}+\dots+c_n\vec{s}_n} of other 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 } , and subtracting 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}_i} from both sides of that equation gives a linear relationship involving a nonzero coefficient, namely 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 -1 } in front 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 \vec{s}_i } .

Example 8

In the vector space of two-wide row vectors, the two-element set 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{pmatrix} 40 &15 \end{pmatrix},\begin{pmatrix} -50 &25 \end{pmatrix}\} } is linearly independent. To check this, set

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\cdot\begin{pmatrix} 40 &15 \end{pmatrix}+c_2\cdot\begin{pmatrix} -50 &25 \end{pmatrix}=\begin{pmatrix} 0 &0 \end{pmatrix} }

and solving the resulting system

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}{*{2}{rc}r} 40c_1 &- &50c_2 &= &0 \\ 15c_1 &+ &25c_2 &= &0 \end{array} \;\xrightarrow[]{-(15/40)\rho_1+\rho_2}\; \begin{array}{*{2}{rc}r} 40c_1 &- &50c_2 &= &0 \\ & &(175/4)c_2 &= &0 \end{array} }

shows that both 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 } are zero. So the only linear relationship between the two given row vectors is the trivial relationship.

In the same 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 \{ \begin{pmatrix} 40 &15 \end{pmatrix},\begin{pmatrix} 20 &7.5 \end{pmatrix}\} } is linearly dependent since we can satisfy

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\begin{pmatrix} 40 &15 \end{pmatrix}+c_2\cdot\begin{pmatrix} 20 &7.5 \end{pmatrix}=\begin{pmatrix} 0 &0 \end{pmatrix} }

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 c_1=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=-2 } .

Example 9

The set 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 \{1+x,1-x\} } is linearly independent 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 \mathcal{P}_2 } , the space of quadratic polynomials with real coefficients, 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 0+0x+0x^2 = c_1(1+x)+c_2(1-x) = (c_1+c_2)+(c_1-c_2)x+0x^2 }

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 \begin{array}{rcl} \begin{array}{*{2}{rc}r} c_1 &+ &c_2 &= &0 \\ c_1 &- &c_2 &= &0 \end{array} &\xrightarrow[]{-\rho_1+\rho_2} &\begin{array}{*{2}{rc}r} c_1 &+ &c_2 &= &0 \\ & &2c_2 &= &0 \end{array} \end{array} }

since polynomials are equal only if their coefficients are equal. Thus, the only linear relationship between these two members 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 \mathcal{P}_2} is the trivial one.

Example 10

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 \mathbb{R}^3 } , 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{v}_1=\begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} \quad \vec{v}_2=\begin{pmatrix} 2 \\ 9 \\ 2 \end{pmatrix} \quad \vec{v}_3=\begin{pmatrix} 4 \\ 18 \\ 4 \end{pmatrix} }

the set 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=\{\vec{v}_1,\vec{v}_2,\vec{v}_3\} } is linearly dependent because this is a relationship

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{v}_1 +2\cdot\vec{v}_2 -1\cdot\vec{v}_3 =\vec{0} }

where not all of the scalars are zero (the fact that some of the scalars are zero doesn't matter).

Resources

• Subspaces, Interactive Linear Algebra from Georgia Tech

Licensing

Content obtained and/or adapted from:

• Definition and Examples of Linear Independence, WikiBooks Linear Algebra under a CC BY-SA license
• Subspaces and Spanning Sets, WikiBooks Linear Algebra under a CC BY-SA license
• Subspaces, WikiBooks Linear Algebra under a CC BY-SA license