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benediktmag merged 1 commit into from
Mar 2, 2026
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stæ107g: gamlar lagfæringar á kafla 4 #238

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215 changes: 107 additions & 108 deletions projects/stae107g/kafli04.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -622,7 +622,7 @@ Sýnidæmi: Grunnur fyrir núllrúm
sem gefur

.. math:: \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix}
= \begin{bmatrix} x_3+2x_4+3x_5 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix}
= \begin{bmatrix} x_3+2x_4+3x_5 \\ -2x_3-3x_4-4x_5 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix}
= x_3 \begin{bmatrix} 1 \\ -2 \\ 1 \\ 0 \\ 0 \end{bmatrix}
+ x_4 \begin{bmatrix} 2 \\ -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}
+ x_5 \begin{bmatrix} 3 \\ -4 \\ 0 \\ 0 \\ 1 \end{bmatrix}
Expand Down Expand Up @@ -706,7 +706,7 @@ Sýnidæmi: Grunnur fyrir dálkrúm
af :math:`\ve u_1` og :math:`\ve u_2`. Því er :math:`\col{A} = \text{Span}\{\ve a_1, \ve a_2 \}`.
Við fáum því að

.. math:: \left\{ \begin{bmatrix} 1 \\ 6\\ 11 \end{bmatrix}, \begin{bmatrix} 2 \\ 7 \\ 1 \end{bmatrix} \right\}
.. math:: \left\{ \begin{bmatrix} 1 \\ 6\\ 11 \end{bmatrix}, \begin{bmatrix} 2 \\ 7 \\ 12 \end{bmatrix} \right\}

er grunnur fyrir :math:`\col{A}`.

Expand All @@ -724,110 +724,6 @@ Setning: Tveir eiginleikar grunna

Þetta má orða sem svo: *Grunnur er eins lítið spannmengi og eins stórt línulega óháð mengi í vigurrúmi og mögulegt er*.

Setning: Um tilvist hnita
~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Setning
:class: setning

Gerum ráð fyrir að :math:`\mathcal{B}=\{\ve b_1, \ve b_2, \ldots, \ve b_n\}` sé grunnur fyrir :math:`V`.
Þá gildir fyrir sérhvern vigur :math:`\ve v\in V` að til eru *ótvírætt* ákvarðaðar tölur :math:`c_1, c_2, \ldots, c_n`
þannig að

.. math:: \ve v= c_1\ve b_1 + c_2\ve b_2 + \ldots + c_n\ve b_n.

Þetta má orða á þann hátt að jafnan :math:`x_1\ve b_1 + x_2\ve b_2 + \ldots + x_n\ve b_n=\ve v` hafi nákvæmlega eina lausn, þ.e. fyrir gefinn grunn þá er til nákvæmlega ein leið til að rita gefinn vigur sem línulega samantekt vigranna í grunninum.

Skilgreining: :math:`\mathcal{B}`-hnit
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Skilgreining
:class: skilgreining

Gerum ráð fyrir að :math:`\mathcal{B}=\{\ve b_1, \ve b_2, \ldots, \ve b_n\}` sé grunnur fyrir :math:`V`.
Tölurnar :math:`c_1, c_2, \ldots, c_n` þannig að

.. math:: \ve v= c_1\ve b_1 + c_2\ve b_2 + \ldots + c_n\ve b_n,

kallast **hnit vigursins** :math:`\ve v` **með tilliti til grunsins** :math:`\mathcal{B}` (e. coordinates of :math:`\ve v` relative to the basis :math:`\mathcal{B}`).
Við tölum líka um :math:`\mathcal{B}`-hnit vigursins :math:`\ve v` (e. :math:`\mathcal{B}`-coordinates of :math:`\ve v`).

Skilgreining: Hnitavigur :math:`[\ve x]_{\mathcal{B}}`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Skilgreining
:class: skilgreining

Ef tölurnar :math:`c_1, c_2, \ldots, c_n` eru hnit vigursins :math:`\ve x` með tilliti til grunsins :math:`\mathcal{B}`
þá segjum við að vigurinn

.. math:: [\ve x]_{\mathcal{B}}=\begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix}

sé **hnitavigur** :math:`\ve x` **með tilliti til grunnsins** :math:`\mathcal{B}`
(e. coordinate vector of :math:`x` relative to :math:`\mathcal{B}`, or :math:`\mathcal{B}`-coordinate vector).


Sýnidæmi: Að skipta um grunn
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.. admonition:: Dæmi
:class: daemi

Gefinn er grunnurinn

.. math:: \mathcal{B} = \{\ve b_1, \ve b_2\}= \left\{\begin{bmatrix} -1 \\ 3 \end{bmatrix},
\begin{bmatrix} 2 \\ 2 \end{bmatrix}\right\}

Vigurinn :math:`\ve x` með tillit til :math:`\mathcal{B}` hefur hnitin :math:`[\ve x]_{\mathcal{B}}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}`. Finnið hnit :math:`\ve x` með tillit til venjulega grunnsins :math:`\mathcal{E}=\{\ve e_1, \ve e_2 \}`, þ.e. finnið :math:`[\ve x]_{\mathcal{E}}`.

.. admonition:: Lausn
:class: daemi, dropdown

Fáum

.. math:: [\ve x]_{\mathcal{E}} = \ve x = 1\begin{bmatrix}
-1\\3
\end{bmatrix}+2\begin{bmatrix}
2\\2
\end{bmatrix}= \begin{bmatrix}
-1+4\\3+4
\end{bmatrix}=
\begin{bmatrix}
3\\7
\end{bmatrix}.

Skilgreining: Hnitavörpun
~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Skilgreining
:class: skilgreining

Gerum ráð fyrir að :math:`\mathcal{B}=\{\ve b_1, \ve b_2, \ldots, \ve b_n\}` sé grunnur fyrir vigurrúm :math:`V`. Vörpunin

.. math:: V\rightarrow\R^n; \quad \ve x\mapsto [\ve x]_{\mathcal{B}}

kallast **hnitavörpunin með tilliti til** :math:`\mathcal{B}` (e. coordiante mapping determined by :math:`\mathcal{B}`).


Setning: hnitavörpun er línuleg
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Setning
:class: setning

Látum :math:`V` vera endanlega vítt vigurrúm með grunn :math:`\mathcal{B}=\{\ve b_1, \dots, \ve b_n \}`. Hnitavörpunin

.. math:: V \rightarrow \R^n, \ \ve x \mapsto [\ve x]_{\mathcal{B}}

er línuleg vörpun. Hnitavörpunin er einnig gagntæk

.. admonition:: Athugasemd
:class: athugasemd

- Hnitavörpun :math:`V \rightarrow W` er mikilvægt dæmi um svokallaða *einsmótun* (e. isomorphism), þ.e. gagntæk línuleg vörpun, á milli vigurrúma.

- Um endanlega víð vigurrúm :math:`V` með vídd :math:`n` gildir að :math:`V` er einsmóta :math:`\R^n`.


Vídd og rankur
--------------
Expand Down Expand Up @@ -1090,10 +986,113 @@ Setning: Viðbót við löngu setninguna um andhverfur
**18.** :math:`\dim{(\nul{A})}=0`.


Hnitaskipti
Hnit og hnitaskipti
-----------
Comment on lines +989 to 990

@coderabbitai coderabbitai Bot Mar 2, 2026

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⚠️ Potential issue | 🟠 Major

Fix section title underline length to valid RST heading syntax.

The underline under Hnit og hnitaskipti is too short for the title text, which can cause Sphinx/Docutils heading parsing warnings (and fail strict doc builds). Please extend the hyphen line to at least the title length.

Suggested fix 
 Hnit og hnitaskipti
------------
+-------------------

As per coding guidelines "For reStructuredText files: Check for proper use of RST syntax and Sphinx directives."

📝 Committable suggestion

‼️ IMPORTANT
Carefully review the code before committing. Ensure that it accurately replaces the highlighted code, contains no missing lines, and has no issues with indentation. Thoroughly test & benchmark the code to ensure it meets the requirements.

Suggested change
Hnit og hnitaskipti
-----------
Hnit og hnitaskipti
-------------------
🤖 Prompt for AI Agents 
Verify each finding against the current code and only fix it if needed.

In `@projects/stae107g/kafli04.rst` around lines 989 - 990, The section heading
"Hnit og hnitaskipti" has an underline of hyphens that is shorter than the title
text; update the underline under that heading so the hyphen line length is at
least equal to the number of characters in "Hnit og hnitaskipti" (i.e., extend
the dashed line to match the title length) to conform to valid reStructuredText
heading syntax and avoid Sphinx/Docutils parsing warnings.


Við höfum áður séð hnitaskiptafylkið :math:`\mathcal{P}_{\mathcal{E}\leftarrow\mathcal{B}}` í sýnidæmi :math:`4.3.7.1.`, þegar :math:`\mathcal{P}_\mathcal{E \leftarrow B}[\ve x]_\mathcal{B}=\ve x`. Nú ætlum við að skoða almennari skilgreiningu fyrir hnitaskipti.
Setning: Um tilvist hnita
~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Setning
:class: setning

Gerum ráð fyrir að :math:`\mathcal{B}=\{\ve b_1, \ve b_2, \ldots, \ve b_n\}` sé grunnur fyrir :math:`V`.
Þá gildir fyrir sérhvern vigur :math:`\ve v\in V` að til eru *ótvírætt* ákvarðaðar tölur :math:`c_1, c_2, \ldots, c_n`
þannig að

.. math:: \ve v= c_1\ve b_1 + c_2\ve b_2 + \ldots + c_n\ve b_n.

Þetta má orða á þann hátt að jafnan :math:`x_1\ve b_1 + x_2\ve b_2 + \ldots + x_n\ve b_n=\ve v` hafi nákvæmlega eina lausn, þ.e. fyrir gefinn grunn þá er til nákvæmlega ein leið til að rita gefinn vigur sem línulega samantekt vigranna í grunninum.

Skilgreining: :math:`\mathcal{B}`-hnit
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Skilgreining
:class: skilgreining

Gerum ráð fyrir að :math:`\mathcal{B}=\{\ve b_1, \ve b_2, \ldots, \ve b_n\}` sé grunnur fyrir :math:`V`.
Tölurnar :math:`c_1, c_2, \ldots, c_n` þannig að

.. math:: \ve v= c_1\ve b_1 + c_2\ve b_2 + \ldots + c_n\ve b_n,

kallast **hnit vigursins** :math:`\ve v` **með tilliti til grunsins** :math:`\mathcal{B}` (e. coordinates of :math:`\ve v` relative to the basis :math:`\mathcal{B}`).
Við tölum líka um :math:`\mathcal{B}`-hnit vigursins :math:`\ve v` (e. :math:`\mathcal{B}`-coordinates of :math:`\ve v`).

Skilgreining: Hnitavigur :math:`[\ve x]_{\mathcal{B}}`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Skilgreining
:class: skilgreining

Ef tölurnar :math:`c_1, c_2, \ldots, c_n` eru hnit vigursins :math:`\ve x` með tilliti til grunsins :math:`\mathcal{B}`
þá segjum við að vigurinn

.. math:: [\ve x]_{\mathcal{B}}=\begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix}

sé **hnitavigur** :math:`\ve x` **með tilliti til grunnsins** :math:`\mathcal{B}`
(e. coordinate vector of :math:`x` relative to :math:`\mathcal{B}`, or :math:`\mathcal{B}`-coordinate vector).


Skilgreining: Hnitavörpun
~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Skilgreining
:class: skilgreining

Gerum ráð fyrir að :math:`\mathcal{B}=\{\ve b_1, \ve b_2, \ldots, \ve b_n\}` sé grunnur fyrir vigurrúm :math:`V`. Vörpunin

.. math:: V\rightarrow\R^n; \quad \ve x\mapsto [\ve x]_{\mathcal{B}}

kallast **hnitavörpunin með tilliti til** :math:`\mathcal{B}` (e. coordiante mapping determined by :math:`\mathcal{B}`).

@coderabbitai coderabbitai Bot Mar 2, 2026

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⚠️ Potential issue | 🟡 Minor

Minor typo in glossary text (coordiante → coordinate).

Please correct the spelling in the English parenthetical for clarity and polish in student-facing material.

Suggested fix 
-    kallast **hnitavörpunin með tilliti til** :math:`\mathcal{B}` (e. coordiante mapping determined by :math:`\mathcal{B}`).
+    kallast **hnitavörpunin með tilliti til** :math:`\mathcal{B}` (e. coordinate mapping determined by :math:`\mathcal{B}`).

As per coding guidelines "The content is free of spelling and grammatical errors."

📝 Committable suggestion

‼️ IMPORTANT
Carefully review the code before committing. Ensure that it accurately replaces the highlighted code, contains no missing lines, and has no issues with indentation. Thoroughly test & benchmark the code to ensure it meets the requirements.

Suggested change
kallast **hnitavörpunin með tilliti til** :math:`\mathcal{B}` (e. coordiante mapping determined by :math:`\mathcal{B}`).
kallast **hnitavörpunin með tilliti til** :math:`\mathcal{B}` (e. coordinate mapping determined by :math:`\mathcal{B}`).
🤖 Prompt for AI Agents 
Verify each finding against the current code and only fix it if needed.

In `@projects/stae107g/kafli04.rst` at line 1045, Fix the typo in the English
parenthetical of the glossary sentence by replacing the misspelled word
"coordiante" with "coordinate" in the phrase "coordiante mapping determined by
:math:`\mathcal{B}`" so it reads "coordinate mapping determined by
:math:`\mathcal{B}`; update the text around that fragment to preserve spacing
and punctuation in the same sentence.


Sýnidæmi: Að skipta um grunn
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.. admonition:: Dæmi
:class: daemi

Gefinn er grunnurinn

.. math:: \mathcal{B} = \{\ve b_1, \ve b_2\}= \left\{\begin{bmatrix} -1 \\ 3 \end{bmatrix},
\begin{bmatrix} 2 \\ 2 \end{bmatrix}\right\}

Vigurinn :math:`\ve x` með tillit til :math:`\mathcal{B}` hefur hnitin :math:`[\ve x]_{\mathcal{B}}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}`. Finnið hnit :math:`\ve x` með tillit til venjulega grunnsins :math:`\mathcal{E}=\{\ve e_1, \ve e_2 \}`, þ.e. finnið :math:`[\ve x]_{\mathcal{E}}`.

.. admonition:: Lausn
:class: daemi, dropdown

Fáum

.. math:: [\ve x]_{\mathcal{E}} = \ve x = 1\begin{bmatrix}
-1\\3
\end{bmatrix}+2\begin{bmatrix}
2\\2
\end{bmatrix}= \begin{bmatrix}
-1+4\\3+4
\end{bmatrix}=
\begin{bmatrix}
3\\7
\end{bmatrix}.


Setning: hnitavörpun er línuleg
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. admonition:: Setning
:class: setning

Látum :math:`V` vera endanlega vítt vigurrúm með grunn :math:`\mathcal{B}=\{\ve b_1, \dots, \ve b_n \}`. Hnitavörpunin

.. math:: V \rightarrow \R^n, \ \ve x \mapsto [\ve x]_{\mathcal{B}}

er línuleg vörpun. Hnitavörpunin er einnig gagntæk

.. admonition:: Athugasemd
:class: athugasemd

- Hnitavörpun :math:`V \rightarrow W` er mikilvægt dæmi um svokallaða *einsmótun* (e. isomorphism), þ.e. gagntæk línuleg vörpun, á milli vigurrúma.

- Um endanlega víð vigurrúm :math:`V` með vídd :math:`n` gildir að :math:`V` er einsmóta :math:`\R^n`.



Skilgreining: Hnitaskiptafylki
Expand Down
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