Examples:: Nonexamples:: *[[Nonexamples]]* Specializations::[[inner product space|inner product space]], [[normed vector space]], [[Hilbert space]], Constructions:: [[linear subspace]] Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* ---- > [!definition] Definition. ([[vector space]]) >A $k$-[[module]], $k$ a [[field]], is called a **vector space over $k$**. > $k$-[[vector space|vector spaces]] are objects of [[category]] $k$-$\mathsf{Vect}$ ($k$-$\mathsf{Vect}^{f}$ if we restrict attention to finite-dimensional spaces). The homset $\text{Hom}_{k\text{-}\mathsf{Vect}}(V,W)$ is the set of [[linear map|linear maps]] between $V$ and $W$, [[vector space of linear maps between two vector spaces|itself a vector space]] (vector space hom). > [!note] Explicitly. > Explicitly: let $V$ be a set of objects called [[vector|vectors]]. Equip $V$ with a [[binary operation]] *vector addition* such that the sum of the vectors $x \text{ and } y$ is a vector denoted $x+y$. Equip $V$ with an operation *scalar multiplication* such that the product of the scalar $c \in k$ and the vector $V$ is a vector denoted $cx$. > \ > The set $V$, along with these two operations, is called a [[vector space]] if the following properties hold — > 1. $x+y = y+x \text{ (commutativity)}$ > 2. $x + (y + z) = (x+y) + z \text{ (additive associativity)}$ > 3. There is a unique vector $0$ such that $x+0 = x$ for all $x$. > 4. $x+(-1)x=0 \ (\text{additive inverse})$ > 5. $1x=x \text{ (multiplicative identity)}$ > 6. $c(dx)=cd(x) \text{(multiplicative associativity)}$ > 7. $(c+d)(x) = cx+dx$ > 8. $c(x+y)=cx+cy$ ^note ---- #### > [!basicexample]- > - The set $\mathbb{R}^n$ of all $n-$tuples of real numbers with component-wise addition of and multiplication by scalars in $\mathbb{R}$ forms a vector space. > - Likewise, $\mathbb{C}^{n}$ forms a [[vector space]]. ---- #### 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 > ```