> [!NOTE] Definition (Vector space) > A vector space over of [[Field (Algebra)|field]] $\mathbb{F}$ is an [[Algebraic Structure|algebraic structure]] $(V,+,f)$ consisting of a non-empty set $V$ together with a [[Binary Operation|binary operation]] $+$ (called addition) and a [[Binary Function|binary function]] (called scalar multiplication) such that $(V,+)$ is an [[Abelian Group|abelian group]] that is closed under scalar multiplication by elements of $\mathbb{F}$: for any $v\in V$ and any $\lambda\in\mathbb{F}$, there is another element $\lambda v\in V$, and this multiplication satisfies the rules: > 1. $\lambda(v+w)=\lambda v+\lambda w$ > 2. $(\lambda+\mu)v=\lambda v+\mu w$ (scalar multiplication is [[Distributivity|distributive]] over addition). > 3. $\lambda(uv)=(\lambda \mu)v$ > 4. $1v=v$ for any $\lambda,\mu\in\mathbb{R}$ and $v,w\in V$. > > We usually denote the additive identity by $0_{V}$ and for any $v\in V$ its additive inverse is $-v.$ > [!Example] > The [[Ring of Polynomial Forms|ring of univariate polynomials with real coefficients in the indeterminate]] $x$ denoted $\mathbb{R}[x]$ is a vector space. > [!Example] > The set of functions $V = \left\{ f: \mathbb{R} \to \mathbb{R} : f \text{ is twice continuously differentiable and } \frac{d^{2}f}{dx^{2}}+9f = 0 \right\}$is a vector space. # Properties > [!NOTE] Additive identities & inverses > Let $V$ be a vector space. For any $\lambda\in\mathbb{R}$ and $v\in V$ we have > 1. $\lambda 0_{V}=0_{V}$ and $0v=0_{V}$ > 2. $(-1)v=-v$ and $(-\lambda)v=-(\lambda v)=\lambda(-v)$. > > >*Proof*. >1. $v+0_{V} =v\implies\lambda(v+0_{V})=\lambda v\implies\lambda v + \lambda 0_{V}=\lambda v\implies\lambda0_{V}=0_{V}$. >2. Note that $\lambda v+(-\lambda)v=(\lambda+(-\lambda))v=0v=0_{V}$ so $-\lambda v=(-\lambda) v$ by uniqueness of inverses. > [!info] Vector Subspace > A [[Vector Subspace|vector subspace]] is a subset of $V$ that is closed under linear combinations. > > [!info] Span > A [[Span of Subset of Vector Space|span]] of a non-empty subset of $V$ is the set that contains all linear combinations of its elements. Note that a span is therefore a vector subspace of $V$. > [!info ] Finite Dimensional > A vector space $V$ is [[Finite Dimensional Vector Space|finite dimensional]] iff there is a finite subset $\{ v_{1},\dots,v_{s} \}\subset V$ that spans $V$ i.e. $\langle v_{1},\dots,v_{s} \rangle=V$. > > Note that every FDVS is [[Linear Isomorphism|linearly isomorphic]] to $\mathbb{R}^{n}$. > > [!info] New spaces from old > See [[Direct sum of vector spaces]]. > [!NOTE] Definition (Inner Product) > An [[Inner Product|inner product]] on a vector space $V$ associates a scalar, denoted $\langle v,w \rangle$ to any $v,w\in V$ subject to the following rules: > 1. Commutativity: $\langle v,w\rangle = \langle w,v\rangle$ for any $v,w\in V.$ > 2. Bilinearity: $\langle (\lambda_{1}v_{1},\lambda_{2}v_{2}),w\rangle = \lambda_{1} \langle v_{1},w\rangle+ \lambda_{2}\langle v_{2},w\rangle$ for any $v_{1},v_{2},w\in V$ and $\lambda_{1}, \lambda_{2} \in \mathbb{F}.$ > 3. Positive Definite: For any $v\in V,$ $\langle v,v\rangle \geq 0,$ and furthermore $\langle v,v\rangle=0$ iff $v=0_{V}.$