Ivan Shishkin, Birch Grove

Scalar

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Revision 2496

8/28/2026, 4:30:50 PM · Catalpa

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Revision 2495

8/28/2026, 4:30:48 PM · Catalpa

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1##### Intuition
2A scalar is a number by which one can multiply a [[vecteur|vector]]. Most of the time, scalars are simply real or complex numbers.
3![image](https://s3.pub2.infomaniak.cloud/object/v1/AUTH_7cc517879b0040959f7d12abb1f0e72d/mathwoods-images/uploads/2026/08/user-1/1787430077665-cae9e95214c94964-image.png#mw-width-43)
4*A vector $v$ multiplied by the scalar $2$.*
5
6
7##### Formal definition
8Let $E$ be a [[Vector space|vector space]] over a field $K$. The elements of $K$ are called the scalars of $E$.
8Let $E$ be a [[Vector space|vector space]] over a [[Field|field]] $K$. The elements of $K$ are called the scalars of $E$.
9Scalar multiplication is the map
10$$K \times E \longrightarrow E, \quad(\lambda, v) \longmapsto \lambda v.$$
11
12
13##### Remarks
14- More generally, if $M$ is a [[module|module]] over a ring $A$, the elements of $A$ may also be called scalars.

Revision 2072

8/22/2026, 8:24:00 PM · Ancient Tree

Concept created

##### Intuition
A scalar is a number by which one can multiply a [[vecteur|vector]]. Most of the time, scalars are simply real or complex numbers.
![image](https://s3.pub2.infomaniak.cloud/object/v1/AUTH_7cc517879b0040959f7d12abb1f0e72d/mathwoods-images/uploads/2026/08/user-1/1787430077665-cae9e95214c94964-image.png#mw-width-43)
*A vector $v$ multiplied by the scalar $2$.*


##### Formal definition
Let $E$ be a [[Vector space|vector space]] over a field $K$. The elements of $K$ are called the scalars of $E$.
Scalar multiplication is the map
$$K \times E \longrightarrow E, \quad(\lambda, v) \longmapsto \lambda v.$$


##### Remarks
- More generally, if $M$ is a [[module|module]] over a ring $A$, the elements of $A$ may also be called scalars.