Examples:: *[[Examples]]*
Nonexamples:: *[[Nonexamples]]*
Constructions:: *[[Constructions|Used in the construction of...]]*
Generalizations:: *[[Generalizations]]*
Justifications and Intuition:: [[orientation-preserving and orientation-reversing operators#^58614e|see here]]
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> [!definition] Definition. ([[orientation-preserving and orientation-reversing operators]])
> Let $h \in \endo(\rrn)$ be the [[linear map]] obtain via multiplication by a [[matrix]] $C \in \rr ^{n \times n}$, $h(x)=Cx$. If $\det C >0$ we say $h$ is **orientation-preserving**; if $\det C < 0$ we say $h$ is **orientation-reversing**.
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> [!justification] Motivation.
> From [[determinant sign determines orientation]] we see that for some $n$-frame $(a_{1}, \dots, a_{n})$ in $\rrn$ and [[linear operator|linear operator]] $h \in \endo(\rrn)$ that $\det h > 0$ implies $(h(a_{1}), \dots, h(a_{n}))$ belong to the same [[orientation of euclidian space]] as $(a_{1},\dots,a_{n})$.
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Similarly, $\det h < 0$ implies $(h(a_{1}), \dots, h(a_{n}))$ belongs to the opposite [[orientation of euclidian space]] as $(a_{1},\dots,a_{n})$.
^58614e
> [!intuition] Intuition.
> This gives us an interpretation for the *sign* of the [[determinant]] – if negative, space has 'flipped around' in some way during the transformation. See here: <iframe title="The determinant | Chapter 6, Essence of linear algebra" src="https://www.youtube.com/embed/Ip3X9LOh2dk?start=225&feature=oembed" height="113" width="200" allowfullscreen="" allow="fullscreen" style="aspect-ratio: 1.76991 / 1; width: 100%; height: 100%;"></iframe>
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#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```