Difference between revisions of "Orthonormal Bases and the Gram-Schmidt Process"
(Created page with "The prior subsection suggests that projecting onto the line spanned by <math> \vec{s} </math> decomposes a vector <math>\vec{v}</math> into two parts <center> <TABLE border=0p...") |
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We will now develop that suggestion. | We will now develop that suggestion. | ||
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Definition 2.1''' | |
| − | Vectors <math> \vec{v}_1,\dots,\vec{v}_k\in\mathbb{R}^n </math> are '''mutually orthogonal''' when any two are orthogonal: if <math> i\neq j </math> then the dot product <math> \vec{v}_i\cdot\vec{v}_j </math> is zero. | + | :Vectors <math> \vec{v}_1,\dots,\vec{v}_k\in\mathbb{R}^n </math> are '''mutually orthogonal''' when any two are orthogonal: if <math> i\neq j </math> then the dot product <math> \vec{v}_i\cdot\vec{v}_j </math> is zero. |
| − | + | </blockquote> | |
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Theorem 2.2''' | |
| − | If the vectors in a set <math> \{\vec{v}_1,\dots,\vec{v}_k\}\subset\mathbb{R}^n </math> are mutually orthogonal and nonzero then that set is linearly independent. | + | : If the vectors in a set <math> \{\vec{v}_1,\dots,\vec{v}_k\}\subset\mathbb{R}^n </math> are mutually orthogonal and nonzero then that set is linearly independent. |
| − | + | </blockquote> | |
| − | + | Proof: | |
| − | |||
Consider a linear relationship | Consider a linear relationship | ||
<math> c_1\vec{v}_1+c_2\vec{v}_2+\dots+c_k\vec{v}_k=\vec{0} </math>. | <math> c_1\vec{v}_1+c_2\vec{v}_2+\dots+c_k\vec{v}_k=\vec{0} </math>. | ||
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shows, since <math> \vec{v}_i </math> is nonzero, that <math> c_i </math> is zero. | shows, since <math> \vec{v}_i </math> is nonzero, that <math> c_i </math> is zero. | ||
| − | |||
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Corollary 2.3''' | |
| − | If the vectors in a size <math> k </math> subset of a <math>k</math> dimensional space are mutually orthogonal and nonzero then that set is a basis for the space. | + | : If the vectors in a size <math> k </math> subset of a <math>k</math> dimensional space are mutually orthogonal and nonzero then that set is a basis for the space. |
| − | + | </blockquote> | |
| − | + | Proof: | |
| − | |||
Any linearly independent size <math> k </math> subset of a <math>k</math> dimensional space is a basis. | Any linearly independent size <math> k </math> subset of a <math>k</math> dimensional space is a basis. | ||
| − | |||
| − | Of course, the converse of | + | Of course, the converse of Corollary 2.3 |
does not hold— not every basis of every subspace | does not hold— not every basis of every subspace | ||
of <math>\mathbb{R}^n</math> is made of mutually orthogonal vectors. | of <math>\mathbb{R}^n</math> is made of mutually orthogonal vectors. | ||
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consisting of mutually orthogonal vectors. | consisting of mutually orthogonal vectors. | ||
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Example 2.4''': | |
The members <math>\vec{\beta}_1</math> and <math>\vec{\beta}_2</math> of this basis for <math>\mathbb{R}^2</math> | The members <math>\vec{\beta}_1</math> and <math>\vec{\beta}_2</math> of this basis for <math>\mathbb{R}^2</math> | ||
are not orthogonal. | are not orthogonal. | ||
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</center> | </center> | ||
which leaves the part, <math>\vec{\kappa}_2</math> pictured above, of <math>\vec{\beta}_2</math> that is orthogonal to <math>\vec{\kappa}_1</math> (it is orthogonal by the definition of the projection onto the span of <math>\vec{\kappa}_1</math>). Note that, by the corollary, <math>\{\vec{\kappa}_1,\vec{\kappa}_2\}</math> is a basis for <math>\mathbb{R}^2</math>. | which leaves the part, <math>\vec{\kappa}_2</math> pictured above, of <math>\vec{\beta}_2</math> that is orthogonal to <math>\vec{\kappa}_1</math> (it is orthogonal by the definition of the projection onto the span of <math>\vec{\kappa}_1</math>). Note that, by the corollary, <math>\{\vec{\kappa}_1,\vec{\kappa}_2\}</math> is a basis for <math>\mathbb{R}^2</math>. | ||
| − | + | </blockquote> | |
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Definition 2.5''': | |
| − | An '''orthogonal basis''' for a vector space is a basis of mutually orthogonal vectors. | + | :An '''orthogonal basis''' for a vector space is a basis of mutually orthogonal vectors. |
| − | + | </blockquote> | |
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Example 2.6''': | |
To turn this basis for <math> \mathbb{R}^3 </math> | To turn this basis for <math> \mathbb{R}^3 </math> | ||
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is a basis for the space. | is a basis for the space. | ||
| − | + | </blockquote> | |
The next result verifies that | The next result verifies that | ||
| Line 159: | Line 155: | ||
definition of orthogonality for other vector spaces). | definition of orthogonality for other vector spaces). | ||
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Theorem 2.7 (Gram-Schmidt orthogonalization)''': | |
| − | If <math> \left\langle \vec{\beta}_1,\ldots\vec{\beta}_k \right\rangle </math> | + | :If <math> \left\langle \vec{\beta}_1,\ldots\vec{\beta}_k \right\rangle </math> |
is a basis for a subspace of <math> \mathbb{R}^n </math> then, where | is a basis for a subspace of <math> \mathbb{R}^n </math> then, where | ||
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the <math> \vec{\kappa}\, </math>'s form an orthogonal basis for the same subspace. | the <math> \vec{\kappa}\, </math>'s form an orthogonal basis for the same subspace. | ||
| − | + | </blockquote> | |
| − | + | Proof: | |
| − | |||
We will use induction to check that each <math> \vec{\kappa}_i </math> is nonzero, | We will use induction to check that each <math> \vec{\kappa}_i </math> is nonzero, | ||
is in the span of <math>\left\langle \vec{\beta}_1,\ldots\vec{\beta}_i \right\rangle </math> | is in the span of <math>\left\langle \vec{\beta}_1,\ldots\vec{\beta}_i \right\rangle </math> | ||
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The check that <math>\vec{\kappa}_3</math> is also | The check that <math>\vec{\kappa}_3</math> is also | ||
orthogonal to the other preceding vector <math>\vec{\kappa}_2</math> is similar. | orthogonal to the other preceding vector <math>\vec{\kappa}_2</math> is similar. | ||
| − | |||
| − | + | Beyond having the vectors in the basis be orthogonal, we can do more; we can arrange for each vector to have length one by dividing each by its own length (we can '''normalize''' the lengths). | |
| − | + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | |
| − | + | '''Example 2.8''': | |
| − | Normalizing the length of each vector in the orthogonal basis of | + | Normalizing the length of each vector in the orthogonal basis of Example 2.6 produces this '''orthonormal basis'''. |
| − | |||
| − | produces this '''orthonormal basis'''. | ||
:<math> | :<math> | ||
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\right\rangle | \right\rangle | ||
</math> | </math> | ||
| − | + | </blockquote> | |
Besides its intuitive appeal, and its analogy with the | Besides its intuitive appeal, and its analogy with the | ||
Revision as of 13:56, 8 October 2021
The prior subsection suggests that projecting onto the line spanned by 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} } decomposes a vector into two parts
|
![{\displaystyle \displaystyle {\vec {v}}={\mbox{proj}}_{[{\vec {s}}\,]}({\vec {v}})\,+\,\left({\vec {v}}-{\mbox{proj}}_{[{\vec {s}}\,]}({\vec {v}})\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0da7af968d921768d5eef2f763959f0628fe2b2b)
that are orthogonal and so are "not interacting". We will now develop that suggestion.
Definition 2.1
- Vectors are mutually orthogonal when any two are orthogonal: if then the dot product 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}_i\cdot\vec{v}_j } is zero.
Theorem 2.2
- If the vectors in a 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 \{\vec{v}_1,\dots,\vec{v}_k\}\subset\mathbb{R}^n } are mutually orthogonal and nonzero then that set is linearly independent.
Proof: Consider a linear 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 c_1\vec{v}_1+c_2\vec{v}_2+\dots+c_k\vec{v}_k=\vec{0} } . 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 i\in [1..k]} then taking the dot product 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{v}_i } with both sides of the equation
- 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}{rl} \vec{v}_i\cdot (c_1\vec{v}_1+c_2\vec{v}_2+\dots+c_k\vec{v}_k) &=\vec{v}_i\cdot\vec{0} \\ c_i\cdot(\vec{v}_i\cdot\vec{v}_i) &=0 \end{array}}
shows, 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}_i } is nonzero, 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 c_i } is zero.
Corollary 2.3
- If the vectors in a size 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 k } subset of 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 k} dimensional space are mutually orthogonal and nonzero then that set is a basis for the space.
Proof: Any linearly independent size 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 k } subset of 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 k} dimensional space is a basis.
Of course, the converse of Corollary 2.3 does not hold— not every basis of every 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}^n} is made of mutually orthogonal vectors. However, we can get the partial converse that for every 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}^n} there is at least one basis consisting of mutually orthogonal vectors.
Example 2.4: The members 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{\beta}_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 \vec{\beta}_2} of this basis 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 \mathbb{R}^2} are not orthogonal.
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 \displaystyle B=\left\langle \begin{pmatrix} 4 \\ 2 \end{pmatrix}, \begin{pmatrix} 1 \\ 3 \end{pmatrix} \right\rangle } 
However, we can derive 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 B} a new basis for the same space that does have mutually orthogonal members. For the first member of the new basis we simply use 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{\beta}_1} .
- 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{\kappa}_1=\begin{pmatrix} 4 \\ 2 \end{pmatrix} }
For the second member of the new basis, we take away 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{\beta}_2} its part in the direction 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{\kappa}_1} ,
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 \displaystyle \vec{\kappa}_2=\begin{pmatrix} 1 \\ 3 \end{pmatrix} -\mbox{proj}_{\scriptstyle[\vec{\kappa}_1]}({\begin{pmatrix} 1 \\ 3 \end{pmatrix}}) =\begin{pmatrix} 1 \\ 3 \end{pmatrix}-\begin{pmatrix} 2 \\ 1 \end{pmatrix}=\begin{pmatrix} -1 \\ 2 \end{pmatrix}} 
which leaves the part, 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{\kappa}_2} pictured above, 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{\beta}_2} that is orthogonal to 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{\kappa}_1} (it is orthogonal by the definition of the projection onto the span 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{\kappa}_1} ). Note that, by the corollary, 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{\kappa}_1,\vec{\kappa}_2\}} is a basis 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 \mathbb{R}^2} .
Definition 2.5:
- An orthogonal basis for a vector space is a basis of mutually orthogonal vectors.
Example 2.6: To turn this basis 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 \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 \left\langle \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} \right\rangle }
into an orthogonal basis, we take the first vector as it is given.
- 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{\kappa}_1=\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} }
We get 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{\kappa}_2 } by starting with the given second 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{\beta}_2} and subtracting away the part of it in the direction 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{\kappa}_1 } .
- 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{\kappa}_2=\begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix} -\mbox{proj}_{[\vec{\kappa}_1]}({\begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}}) =\begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}-\begin{pmatrix} 2/3 \\ 2/3 \\ 2/3 \end{pmatrix} =\begin{pmatrix} -2/3 \\ 4/3 \\ -2/3 \end{pmatrix} }
Finally, we get 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{\kappa}_3} by taking the third given vector and subtracting the part of it in the direction 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{\kappa}_1 } , and also the part of it in the direction 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{\kappa}_2 } .
- 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{\kappa}_3=\begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} -\mbox{proj}_{[\vec{\kappa}_1]}({\begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}}) -\mbox{proj}_{[\vec{\kappa}_2]}({\begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}}) =\begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} }
Again the corollary 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 \left\langle \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -2/3 \\ 4/3 \\ -2/3 \end{pmatrix}, \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} \right\rangle }
is a basis for the space.
The next result verifies that the process used in those examples works with any basis for any subspace of an 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} (we are restricted to 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} only because we have not given a definition of orthogonality for other vector spaces).
Theorem 2.7 (Gram-Schmidt orthogonalization):
- 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 \left\langle \vec{\beta}_1,\ldots\vec{\beta}_k \right\rangle }
is a basis for a subspace of then, where
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{\kappa}\, } 's form an orthogonal basis for the same subspace.
![{\displaystyle {\begin{array}{rl}{\vec {\kappa }}_{1}&={\vec {\beta }}_{1}\\{\vec {\kappa }}_{2}&={\vec {\beta }}_{2}-{\mbox{proj}}_{[{\vec {\kappa }}_{1}]}({{\vec {\beta }}_{2}})\\{\vec {\kappa }}_{3}&={\vec {\beta }}_{3}-{\mbox{proj}}_{[{\vec {\kappa }}_{1}]}({{\vec {\beta }}_{3}})-{\mbox{proj}}_{[{\vec {\kappa }}_{2}]}({{\vec {\beta }}_{3}})\\&\vdots \\{\vec {\kappa }}_{k}&={\vec {\beta }}_{k}-{\mbox{proj}}_{[{\vec {\kappa }}_{1}]}({{\vec {\beta }}_{k}})-\cdots -{\mbox{proj}}_{[{\vec {\kappa }}_{k-1}]}({{\vec {\beta }}_{k}})\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b81f989d7eaa8606b764dc1db82f5e8c753798)
Proof: We will use induction to check that each 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{\kappa}_i } is nonzero, is in the span 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 \left\langle \vec{\beta}_1,\ldots\vec{\beta}_i \right\rangle } and is orthogonal to all preceding 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{\kappa}_1\cdot\vec{\kappa}_i= \cdots =\vec{\kappa}_{i-1}\cdot\vec{\kappa}_{i}=0 } . With those, and with Corollary 2.3, we will have 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 \left\langle \vec{\kappa}_1,\ldots\vec{\kappa}_k \right\rangle } is a basis for the same space 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 \left\langle \vec{\beta}_1,\ldots\vec{\beta}_k \right\rangle } .
We shall cover the cases up to 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 i=3 } , which give the sense of the argument. Completing the details is Problem 15.
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 i=1 } case is trivial— setting 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{\kappa}_1 } equal to 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{\beta}_1 } makes it a nonzero vector 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{\beta}_1} is a member of a basis, it is obviously in the desired span, and the "orthogonal to all preceding vectors" condition is vacuously met.
For 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 i=2 } case, expand the definition 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{\kappa}_2} .
- 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{\kappa}_2=\vec{\beta}_2 -\mbox{proj}_{[\vec{\kappa}_1]}({\vec{\beta}_2})= \vec{\beta}_2 -\frac{\vec{\beta}_2\cdot\vec{\kappa}_1}{ \vec{\kappa}_1\cdot\vec{\kappa}_1} \cdot\vec{\kappa}_1 = \vec{\beta}_2 -\frac{\vec{\beta}_2\cdot\vec{\kappa}_1}{ \vec{\kappa}_1\cdot\vec{\kappa}_1} \cdot\vec{\beta}_1 }
This expansion shows 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{\kappa}_2} is nonzero or else this would be a non-trivial linear dependence among 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{\beta} } 's (it is nontrivial because the coefficient 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{\beta}_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 1} ) and also shows 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{\kappa}_2} is in the desired span. Finally, 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{\kappa}_2} is orthogonal to the only preceding 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{\kappa}_1\cdot\vec{\kappa}_2= \vec{\kappa}_1\cdot(\vec{\beta}_2 -\mbox{proj}_{[\vec{\kappa}_1]}({\vec{\beta}_2}))=0 }
because this projection is orthogonal.
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 i=3 } case is the same as 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 i=2} case except for one detail. As in 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 i=2} case, expanding the definition
- 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}{rl} \vec{\kappa}_3 &=\vec{\beta}_3 -\frac{\vec{\beta}_3\cdot\vec{\kappa}_1}{ \vec{\kappa}_1\cdot\vec{\kappa}_1} \cdot\vec{\kappa}_1 -\frac{\vec{\beta}_3\cdot\vec{\kappa}_2}{ \vec{\kappa}_2\cdot\vec{\kappa}_2} \cdot\vec{\kappa}_2 \\ &=\vec{\beta}_3 -\frac{\vec{\beta}_3\cdot\vec{\kappa}_1}{ \vec{\kappa}_1\cdot\vec{\kappa}_1} \cdot\vec{\beta}_1 -\frac{\vec{\beta}_3\cdot\vec{\kappa}_2}{ \vec{\kappa}_2\cdot\vec{\kappa}_2} \cdot\bigl(\vec{\beta}_2 -\frac{\vec{\beta}_2\cdot\vec{\kappa}_1}{ \vec{\kappa}_1\cdot\vec{\kappa}_1} \cdot\vec{\beta}_1\bigr) \end{array}}
shows 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{\kappa}_3} is nonzero and is in the span. A calculation shows 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{\kappa}_3} is orthogonal to the preceding 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{\kappa}_1} .
- 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}{rl} \vec{\kappa}_1\cdot\vec{\kappa}_3 &=\vec{\kappa}_1\cdot\bigl(\vec{\beta}_3 -\mbox{proj}_{[\vec{\kappa}_1]}({\vec{\beta}_3}) -\mbox{proj}_{[\vec{\kappa}_2]}({\vec{\beta}_3})\bigr) \\ &=\vec{\kappa}_1\cdot\bigl(\vec{\beta}_3 -\mbox{proj}_{[\vec{\kappa}_1]}({\vec{\beta}_3})\bigr) -\vec{\kappa}_1\cdot\mbox{proj}_{[\vec{\kappa}_2]}({\vec{\beta}_3}) \\ &=0 \end{array}}
(Here's the difference from 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 i=2} case— the second line has two kinds of terms. The first term is zero because this projection is orthogonal, as in 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 i=2} case. The second term is zero 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 \vec{\kappa}_1 } is orthogonal to 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{\kappa}_2 } and so is orthogonal to any vector in the line spanned by 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{\kappa}_2 } .) The check 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{\kappa}_3} is also orthogonal to the other preceding 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{\kappa}_2} is similar.
Beyond having the vectors in the basis be orthogonal, we can do more; we can arrange for each vector to have length one by dividing each by its own length (we can normalize the lengths).
Example 2.8: Normalizing the length of each vector in the orthogonal basis of Example 2.6 produces this orthonormal basis.
- 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 \left\langle \begin{pmatrix} 1/\sqrt{3} \\ 1/\sqrt{3} \\ 1/\sqrt{3} \end{pmatrix}, \begin{pmatrix} -1/\sqrt{6} \\ 2/\sqrt{6} \\ -1/\sqrt{6} \end{pmatrix}, \begin{pmatrix} -1/\sqrt{2} \\ 0 \\ 1/\sqrt{2} \end{pmatrix} \right\rangle }
Besides its intuitive appeal, and its analogy with the standard basis 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{E}_n} 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 \mathbb{R}^n} , an orthonormal basis also simplifies some computations.
Resources
- Gram-Schmidt Orthogonalization, Wikibooks: Linear Algebra
