Difference between revisions of "Subspaces of Rn and Linear Independence"
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We usually show that a subset is a subspace with <math> (2)\implies (1) </math>. | We usually show that a subset is a subspace with <math> (2)\implies (1) </math>. | ||
| − | + | ===Example 1=== | |
: The plane <math> P=\{\begin{pmatrix} x \\ y \\ z \end{pmatrix}\,\big|\, x+y+z=0\} </math> is a subspace of <math> \mathbb{R}^3 </math>. As specified in the definition, the operations are the ones inherited from the larger space, that is, vectors add in <math>P</math> as they add in <math>\mathbb{R}^3</math> | : The plane <math> P=\{\begin{pmatrix} x \\ y \\ z \end{pmatrix}\,\big|\, x+y+z=0\} </math> is a subspace of <math> \mathbb{R}^3 </math>. As specified in the definition, the operations are the ones inherited from the larger space, that is, vectors add in <math>P</math> as they add in <math>\mathbb{R}^3</math> | ||
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: and scalar multiplication is also the same as it is in <math>\mathbb{R}^3</math>. To show that <math>P</math> is a subspace, we need only note that it is a subset and then verify that it is a space. Checking that <math>P</math> 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 <math>x_1+y_1+z_1=0</math> and <math>x_2+y_2+z_2=0</math> then the sum satisfies that <math>(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</math>. | : and scalar multiplication is also the same as it is in <math>\mathbb{R}^3</math>. To show that <math>P</math> is a subspace, we need only note that it is a subset and then verify that it is a space. Checking that <math>P</math> 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 <math>x_1+y_1+z_1=0</math> and <math>x_2+y_2+z_2=0</math> then the sum satisfies that <math>(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</math>. | ||
| − | + | ===Example 2=== | |
: The <math> x </math>-axis in <math> \mathbb{R}^2 </math> is a subspace where the addition and scalar multiplication operations are the inherited ones. | : The <math> x </math>-axis in <math> \mathbb{R}^2 </math> is a subspace where the addition and scalar multiplication operations are the inherited ones. | ||
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: 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. | : 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 <math>\mathbb{R}^2</math> is | : Another subspace of <math>\mathbb{R}^2</math> is | ||
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{{anchor|proper}}Other subspaces are '''proper'''. | {{anchor|proper}}Other 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 <math> \{1\} </math> of the vector space <math> \mathbb{R}^1 </math>. Under the operations <math>1+1=1</math> and <math>r\cdot 1=1</math> that set is a vector space, specifically, a trivial space. But it is not a subspace of <math> \mathbb{R}^1 </math> because those aren't the inherited operations, since of course <math> \mathbb{R}^1 </math> has <math> 1+1=2 </math>. | 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 <math> \{1\} </math> of the vector space <math> \mathbb{R}^1 </math>. Under the operations <math>1+1=1</math> and <math>r\cdot 1=1</math> that set is a vector space, specifically, a trivial space. But it is not a subspace of <math> \mathbb{R}^1 </math> because those aren't the inherited operations, since of course <math> \mathbb{R}^1 </math> has <math> 1+1=2 </math>. | ||
| − | + | ===Example 5=== | |
: All kinds of vector spaces, not just <math>\mathbb{R}^n</math>'s, have subspaces. The vector space of cubic polynomials <math> \{a+bx+cx^2+dx^3\,\big|\, a,b,c,d\in\mathbb{R}\} </math> has a subspace comprised of all linear polynomials <math> \{m+nx\,\big|\, m,n\in\mathbb{R}\} </math>. | : All kinds of vector spaces, not just <math>\mathbb{R}^n</math>'s, have subspaces. The vector space of cubic polynomials <math> \{a+bx+cx^2+dx^3\,\big|\, a,b,c,d\in\mathbb{R}\} </math> has a subspace comprised of all linear polynomials <math> \{m+nx\,\big|\, m,n\in\mathbb{R}\} </math>. | ||
| − | + | ===Example 6=== | |
: This is a subspace of the <math> 2 \! \times \! 2 </math> matrices | : This is a subspace of the <math> 2 \! \times \! 2 </math> matrices | ||
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: As above, we've described the subspace as a collection of unrestricted linear combinations (by coincidence, also of two elements). | : 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 <math> S </math> of a vector space is the set of all linear combinations of vectors from <math> S </math>. | ||
| + | :<math> | ||
| + | [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 \} | ||
| + | </math> | ||
| + | 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 "<math>\mbox{span}(S)</math>" and "<math>\mbox{sp}(S)</math>". | ||
| + | |||
| + | ===Lemma=== | ||
| + | In a vector space, the span of any subset is a subspace. | ||
| + | |||
| + | Proof: | ||
| + | : Call the subset <math> S </math>. If <math> S </math> is empty then by definition its span is the trivial subspace. If <math> S</math> is not empty, then we need only check that the span <math> [S] </math> is closed under linear combinations. For a pair of vectors from that span, <math> \vec{v}=c_1\vec{s}_1+\cdots+c_n\vec{s}_n </math> and <math> \vec{w}=c_{n+1}\vec{s}_{n+1}+\cdots+c_m\vec{s}_m </math>, a linear combination | ||
| + | :: <math> | ||
| + | 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)</math> | ||
| + | ::<math> | ||
| + | = | ||
| + | pc_1\vec{s}_1+\cdots+pc_n\vec{s}_n | ||
| + | +rc_{n+1}\vec{s}_{n+1}+\cdots+rc_m\vec{s}_m | ||
| + | </math> | ||
| + | : (<math> p </math>, <math> r </math> scalars) is a linear combination of elements of <math> S </math> and so is in <math> [S] </math> (possibly some of the <math>\vec{s}_i</math>'s forming <math>\vec{v}</math> equal some of the <math>\vec{s}_j</math>'s from <math>\vec{w}</math>, but it does not matter). | ||
Revision as of 14:58, 29 September 2021
Contents
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 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, under the inherited operations, the following are equivalent statements.
- 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 subspace of that 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 S } is closed under linear combinations of pairs of vectors: for any 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 \vec{s}_1,\vec{s}_2\in S } and 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 r_1,r_2 } the 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 r_1\vec{s}_1+r_2\vec{s}_2 } 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 }
- 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 of any number of vectors: for any 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 \vec{s}_1,\ldots,\vec{s}_n\in S } and 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 r_1, \ldots,r_n } the 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 r_1\vec{s}_1+\cdots+r_n\vec{s}_n } 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 } .
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.
- 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)\!\iff\!(2) \qquad (2)\!\iff\!(3) \qquad (3)\!\iff\!(1) }
- We will show this equivalence by establishing 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 (1)\implies (3)\implies (2)\implies (1)} . This strategy is suggested by noticing 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 (1)\implies (3) } 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 (3)\implies (2) } are easy and so we need only argue the single implication 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) } .
- For that argument, 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 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 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 } , 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 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 } 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 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 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 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 } (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 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 } . 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 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} 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 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 0\cdot\vec{s}+0\cdot\vec{s}=\vec{0} } 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 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 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 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 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{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 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} } 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 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 } . As specified in the definition, the operations are the ones inherited from the larger space, that is, vectors add in as they 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 \mathbb{R}^3}
- 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} 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 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} . 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 P} is a subspace, we need only note that it is a subset and then verify that it is a space. Checking 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 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 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+y_1+z_1=0} 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 x_2+y_2+z_2=0} then the sum satisfies 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 (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 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 }
-axis 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 }
is a subspace where the addition and scalar multiplication operations are the inherited ones.
- 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} 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 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
- 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} 0 \\ 0 \end{pmatrix}\} }
- which is its trivial subspace.
- Any vector space has a trivial subspace 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}\,\} } .
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 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\} } of the 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 \mathbb{R}^1 } . Under the operations 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+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 r\cdot 1=1} that set is a vector space, specifically, a trivial space. But it is not a subspace 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}^1 } because those aren't the inherited operations, since of course 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}^1 } has 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+1=2 } .
Example 5
- All kinds of vector spaces, not just 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}^n} 's, have subspaces. The vector space of cubic polynomials 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+bx+cx^2+dx^3\,\big|\, a,b,c,d\in\mathbb{R}\} } has a subspace comprised of all linear polynomials 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 \{m+nx\,\big|\, m,n\in\mathbb{R}\} } .
Example 6
- This is a subspace 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 2 \! \times \! 2 } matrices
- 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} 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 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 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 "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
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).
