Vector Spaces

Let $\mathbb{X}$ be a set. A binary operation on $\mathbb{X}$ is a function $\mathbb{X} \times \mathbb{X} \rightarrow \mathbb{X}$ that maps each ordered pair of elements of $\mathbb{X}$ to an element of $\mathbb{X}.$

A field $\mathbb{F}$ is a set with two binary operations, called addition $a+b$ and multiplication $a \cdot b,$ that satisfy the field axioms for all $a,b,c \in \mathbb{F}:$

• Associativity of addition and multiplication: $a + (b + c) = (a + b) + c$ and $a \cdot (b \cdot c) = (a \cdot b) \cdot c.$
• Commutativity of addition and multiplication: $a + b = b + a$ and $a \cdot b = b \cdot a.$
• Distributivity of multiplication over addition: $a \cdot (b + c) = (a \cdot b) + (a \cdot c).$
• Additive and multiplicative identity: $\exists \, 0, 1 \in \mathbb{F}$ with $0 \neq 1$ such that $a + 0 = a$ and $a \cdot 1 = a \,\, \forall \, a \in \mathbb{F}.$
• Additive and multiplicative inverse: $\forall \, a \in \mathbb{F} \,\, \exists \, {-}a \in \mathbb{F} : a+(-a)=0$ and $\forall \, a \in \mathbb{F}\setminus \{0\} \,\, \exists \, a^{-1} \in \mathbb{F} : a \cdot a^{-1} = 1.$

Let $\mathbb{F}$ be a field. An element $\lambda \in \mathbb{F}$ is a scalar. The real numbers $\mathbb{R}$ and the complex numbers $\mathbb{C}$ are fields, and usually $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}.$

Let $\mathbb{F}$ be a field and $\mathcal{V}$ be a set with a binary operation called addition $\underline{u} + \underline{v}$ and a function called scalar multiplication $\mathbb{F} \times \mathcal{V} \rightarrow \mathcal{V},$ written $\lambda \cdot \underline{u}.$ The set $\mathcal{V}$ is a vector space over $\mathbb{F}$ if and only if the vector space axioms hold for all $\underline{u}, \underline{v}, \underline{w} \in \mathcal{V}$ and all $\lambda, \mu \in \mathbb{F}:$

• Associativity of addition and scalar multiplication: $\underline{u}+(\underline{v}+\underline{w})=(\underline{u}+\underline{v})+\underline{w}$ and $\lambda\cdot(\mu\cdot\underline{u})=(\lambda\mu)\cdot\underline{u}.$
• Commutativity of addition: $\underline{u}+\underline{v}=\underline{v}+\underline{u}.$
• Distributivity of scalar multiplication over addition: $(\lambda+\mu)\cdot\underline{u}=\lambda\cdot\underline{u}+\mu\cdot\underline{u}$ and $\lambda\cdot(\underline{u}+\underline{v})=\lambda\cdot\underline{u}+\lambda\cdot\underline{v}.$
• Additive identity: $\exists \, \underline{0} \in \mathcal{V} : \underline{u}+\underline{0}=\underline{u}.$
• Scalar multiplicative identity: $1\cdot\underline{u}=\underline{u}$ where $1$ is the multiplicative identity of $\mathbb{F}.$
• Additive inverse: $\exists \, {-}\underline{u} \in \mathcal{V} : \underline{u}+({-}\underline{u})=\underline{0}.$

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F}.$ An element $\underline{v} \in \mathcal{V}$ is a vector. Usually $\mathcal{V}=\mathbb{R}^n$ or $\mathcal{V}=\mathbb{C}^n,$ the set of $n$-tuples $\underline{v}=(v_1,\ldots,v_n)$ with $v_1,\ldots,v_n \in \mathbb{R}$ or $v_1,\ldots,v_n \in \mathbb{C}$ respectively, with element-wise addition and scalar multiplication.

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F}.$ A subspace of $\mathcal{V}$ is a subset $\mathcal{U} \subseteq \mathcal{V}$ that is itself a vector space over $\mathbb{F}$ under the addition and scalar multiplication of $\mathcal{V}.$ A subset of $\mathcal{V}$ that does not satisfy all of the vector space axioms is not a subspace.

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F},$ $\underline{u}_1,\ldots,\underline{u}_n \in \mathcal{V}$ and $\alpha_1,\ldots,\alpha_n \in \mathbb{F}.$ The vector $\alpha_1\underline{u}_1+\ldots+\alpha_n\underline{u}_n$ is a linear combination of $\underline{u}_1,\ldots,\underline{u}_n$ with coefficients $\alpha_1,\ldots,\alpha_n.$

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F}$ and $U=\{\underline{u}_1,\ldots,\underline{u}_n\}$ be a finite subset of $\mathcal{V}.$ The vectors in $U$ are linearly dependent if and only if at least one vector $\underline{u}_k$ is a linear combination of the other vectors in $U.$

$$\underline{u}_k = \sum_{i=1, i\neq k}^n \alpha_i\underline{u}_i$$

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F}$ and $U=\{\underline{u}_1,\ldots,\underline{u}_n\}$ be a finite subset of $\mathcal{V}.$ The vectors in $U$ are linearly independent if and only if they are not linearly dependent. Equivalently, the vectors in $U$ are linearly independent if and only if the only solution to $\alpha_1\underline{u}_1+\ldots+\alpha_n\underline{u}_n=\underline{0}$ is $\alpha_1=\ldots=\alpha_n=0.$ An infinite subset $U$ of $\mathcal{V}$ is linearly independent if and only if every finite subset of $U$ is linearly independent.

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F}$ and $U$ be a subset of $\mathcal{V}.$ The set $U$ spans $\mathcal{V}$ if and only if every vector in $\mathcal{V}$ is a linear combination of vectors in $U.$

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F}$ and $U$ be a subset of $\mathcal{V}.$ The set $U$ is a basis of $\mathcal{V}$ if and only if $U$ spans $\mathcal{V}$ and the vectors in $U$ are linearly independent.

Let $\mathcal{V}$ be a vector space over the field $\mathbb{F}$ with a basis $U.$ The dimension of $\mathcal{V},$ written $\mathrm{dim}(\mathcal{V}),$ is the number of vectors in $U.$ Every vector space spanned by a finite set has a basis, and all bases of a vector space have the same number of vectors, so the dimension of a vector space is uniquely defined. A vector space with a finite basis is finite dimensional.

Let $\mathbb{F}$ be a field, $n \geq 1$ and $i\in\{1,\ldots,n\}.$ The standard basis vector $\underline{e}_i\in\mathbb{F}^n$ is the $n$-tuple with $1$ in position $i$ and $0$ elsewhere. The standard basis of $\mathbb{F}^n$ is the set $\{\underline{e}_1,\ldots,\underline{e}_n\},$ which is a basis of $\mathbb{F}^n,$ so $\mathrm{dim}(\mathbb{F}^n)=n.$