Ivan Shishkin, Birch Grove

Vector space

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Revision 2025

8/22/2026, 2:45:06 PM · Ancient Tree

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1##### Intuition
2A vector space is a set containing [[Vector|vectors]], whose essential properties are that they can be added together and multiplied by [[Scalar|scalars]]. Vectors can be thought of as “arrows,” as they are usually introduced in earlier classes, but one later realizes that other objects can also be regarded as vectors, such as matrices, polynomials, functions, etc.
3![vectorspace](https://s3.pub2.infomaniak.cloud/object/v1/AUTH_7cc517879b0040959f7d12abb1f0e72d/mathwoods-images/uploads/2026/08/user-1/1787409596367-c98709243f77aa65-vectorspace.png#mw-width-95)
4*The addition of vectors and the addition of functions illustrate the same algebraic structure.*
4*The addition of vectors and the addition of functions illustrate the same [[algebraic structure|algebraic structure]].*
5
6The formal definition below ensures that these two operations, addition and scalar multiplication, behave consistently with the usual rules of algebra.
7
8##### Formal definition
9A vector space over a [[Field|field]] $\mathbb{K}$ is a set $E$ equipped with two operations :
10- Vector addition $+:E\times E \rightarrow E$
11- Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$
12
13such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:
141) $E$ is an [[Abelian group|abelian group]] with $+$.
152) Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
163) Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
174) Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
185) Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.

Revision 2024

8/22/2026, 2:44:45 PM · Ancient Tree

Updated text

Compare with revision 20232 changed lines
1##### Intuition
2A vector space is a set containing [[Vector|vectors]], whose essential properties are that they can be added together and multiplied by [[Scalar|scalars]]. Vectors can be thought of as “arrows,” as they are usually introduced in earlier classes, but one later realizes that other objects can also be regarded as vectors, such as matrices, polynomials, functions, etc.
3![vectorspace](https://s3.pub2.infomaniak.cloud/object/v1/AUTH_7cc517879b0040959f7d12abb1f0e72d/mathwoods-images/uploads/2026/08/user-1/1787409596367-c98709243f77aa65-vectorspace.png#mw-width-95)
4*The addition of vectors* ([[relation de Chasles|relation de Chasles]]) *and the addition of functions illustrate the same [[structure algébrique|algebraic structure]].*
4*The addition of vectors and the addition of functions illustrate the same algebraic structure.*
5
6The formal definition below ensures that these two operations, addition and scalar multiplication, behave consistently with the usual rules of algebra.
7
8##### Formal definition
9A vector space over a [[Field|field]] $\mathbb{K}$ is a set $E$ equipped with two operations :
10- Vector addition $+:E\times E \rightarrow E$
11- Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$
12
13such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:
141) $E$ is an [[Abelian group|abelian group]] with $+$.
152) Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
163) Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
174) Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
185) Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.

Revision 2023

8/22/2026, 2:43:15 PM · Ancient Tree

Updated text

Compare with revision 20192 changed lines
1##### Intuition
2A vector space is a set containing [[Vector|vectors]], whose essential properties are that they can be added together and multiplied by [[Scalar|scalars]]. Vectors can be thought of as “arrows,” as they are usually introduced in earlier classes, but one later realizes that other objects can also be regarded as vectors, such as matrices, polynomials, functions, etc.
3![vectorspace](https://s3.pub2.infomaniak.cloud/object/v1/AUTH_7cc517879b0040959f7d12abb1f0e72d/mathwoods-images/uploads/2026/08/user-1/1787409596367-c98709243f77aa65-vectorspace.png#mw-width-95)
4*The addition of vectors* ([[relation de Chasles|relation de Chasles]]) *and the addition of functions illustrate the same [[structure algébrique|algebraic structure]].*
3
4The formal definition below ensures that these two operations, addition and scalar multiplication, behave consistently with the usual rules of algebra.
5
6##### Formal definition
7A vector space over a [[Field|field]] $\mathbb{K}$ is a set $E$ equipped with two operations :
8- Vector addition $+:E\times E \rightarrow E$
9- Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$
10
11such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:
121) $E$ is an [[Abelian group|abelian group]] with $+$.
132) Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
143) Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
154) Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
165) Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.

Revision 2019

8/22/2026, 2:38:40 PM · Ancient Tree

Shared settings updated from fr translation

domainGeneral algebraLinear algebra

Revision 2015

8/22/2026, 1:45:14 PM · Ancient Tree

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Revision 2014

8/22/2026, 1:44:45 PM · Ancient Tree

Updated text

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1##### Intuition
2A vector space is a set containing [[Vector|vectors]], whose essential properties are that they can be added together and multiplied by [[Scalar|scalars]]. Vectors can be thought of as “arrows,” as they are usually introduced in earlier classes, but one later realizes that other objects can also be regarded as vectors, such as matrices, polynomials, functions, etc.
3
4The formal definition below ensures that these two operations, addition and scalar multiplication, behave consistently with the usual rules of algebra.
5
6##### Formal definition
1A vector space over a [[Field|field]] $\mathbb{K}$ is a set $E$ equipped with two operations :
2- Vector addition $+:E\times E \rightarrow E$
3- Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$
4
5such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:
61) $E$ is an [[Abelian group|abelian group]] with $+$.
72) Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
83) Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
94) Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
105) Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.

Revision 254

7/7/2026, 9:26:14 AM · Ancient Tree

Concept edited

Compare with revision 18511 changed lines
1A vector space is...
1A vector space over a [[Field|field]] $\mathbb{K}$ is a set $E$ equipped with two operations :
2- Vector addition $+:E\times E \rightarrow E$
3- Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$
4
5such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:
61) $E$ is an [[Abelian group|abelian group]] with $+$.
72) Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
83) Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
94) Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
105) Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.

Revision 185

7/2/2026, 1:17:44 PM · Ancient Tree

Concept created

A vector space is...