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A positive definite quadratic form is a real quadratic form that takes a positive value at every nonzero vector.

The kernel of a linear map is the set of vectors that the map sends to zero. It is a subspace of the domain.

The dot product of two vectors of $\R^n$ is the sum of the products of their corresponding coordinates.

The external direct sum of a family of vector spaces is the space of families of vectors, one vector from each space. Only finitely many entries of such a family are nonzero. For finitely many spaces it is the product of the spaces.

An orthonormal basis of a finite-dimensional inner product space is a basis whose vectors have length one and are pairwise orthogonal.

A bilinear form on a vector space is a map of two vectors to the scalars. It is linear in each vector when the other vector is fixed.

The augmented matrix of a system of linear equations is the coefficient matrix with the column of constant terms added on the right.

A symmetric matrix is a square matrix that is equal to its transpose.

An orthogonal matrix is a real square matrix whose transpose is its inverse, so that its columns form an orthonormal basis.

A matrix is in reduced row echelon form when it is in row echelon form and each leading entry is $1$. Also, each leading entry is the only nonzero entry in its column.

The image of a linear map is the set of all values of the map. It is a subspace of the codomain.

The space of linear maps between two vector spaces is the set of all linear maps from the first space to the second. Its operations are pointwise addition and pointwise scalar multiplication.

The orthogonal complement of a subspace of an inner product space is the set of all vectors orthogonal to every vector of the subspace.

A vector space is the direct sum of a family of subspaces when each vector is a sum of vectors from the subspaces. This sum must be unique.

The companion matrix of a monic polynomial is a square matrix with ones just below the diagonal. Its last column holds the negated coefficients, and all other entries are zero. Its characteristic polynomial is the given polynomial.

The dual basis of a basis $b_1, \dots, b_n$ is the family of linear forms $f_1, \dots, f_n$ with $f_i(b_j) = 1$ when $i = j$ and $0$ otherwise.

A finite-dimensional vector space is a vector space that has a finite basis, or equivalently a finite spanning set.

Let $G$ be a cyclic group. Then $G$ is abelian.

Let $p$ be a prime number and let $G$ be a group of order $p$. Then $G$ is cyclic, and each element of $G$ other than the identity generates $G$.

Let $G$ be a finite group and let $g \in G$. Then the order of $g$ divides the order $|G|$ of $G$.

Let $G$ and $H$ be groups and let $\varphi \colon G \to H$ be a group homomorphism. Then the kernel $\ker \varphi$ is a normal subgroup of $G$.

Let $G$ and $H$ be groups and let $\varphi \colon G \to H$ be a group homomorphism. Then the map $g \ker \varphi \mapsto \varphi(g)$ is a well-defined isomorphism from the quotient group $G / \ker \varphi$ to the image $\operatorname{im} \varphi$.

Let $G$ be a group with identity element $e$, and let $a, b \in G$. If $ab = e$, then $b = a^{-1}$.

Let $G$ and $H$ be groups, let $e_G$ be the identity element of $G$, and let $\varphi \colon G \to H$ be a group homomorphism. Then $\varphi$ is injective if and only if $\ker \varphi = \{e_G\}$.

Let $G$ be a group and let $H$ be a subgroup of $G$ of index $2$. Then $H$ is normal in $G$.

Let $G$ be a finite group, let $p$ be a prime number, and let $p^n$ be the largest power of $p$ that divides $|G|$. A Sylow $p$-subgroup of $G$ is a subgroup of order $p^n$.

The quotient ring of a ring by a two-sided ideal is the set of the cosets of the ideal. Its operations come from the operations of the ring. The integers modulo $n$ are the main example. The definition of the quotient ring gives the construction.

A prime ideal of a commutative ring is a proper ideal with the following property. If it contains a product, then it contains one of the factors. The prime ideals of $\mathbb{Z}$ are the ideals $p\mathbb{Z}$ for $p$ prime and the zero ideal. The definition of a prime ideal gives the conditions.

A prime element of a commutative ring is a nonzero nonunit $p$ with the following property. If $p$ divides a product, then $p$ divides one of the factors. The definition of a prime element gives the conditions.

A group action is transitive when it has exactly one orbit. Then each point can be moved onto each other point by some element of the group.

The centraliser of a subset $S$ of a group is the set of elements that commute with each element of $S$. It is a subgroup.

A commutative ring is a ring in which the multiplication is commutative. Commutative algebra is the study of these rings. The definition of a commutative ring gives the precise condition.

The symmetric group on a set $X$ is the group of all bijections from $X$ to $X$, with composition as the operation. For $X = \{1, \dots, n\}$ it is written $S_n$, and its elements are the permutations of $n$ symbols.

A submodule of a module is a subset that is a subgroup under addition and that is closed under multiplication by the ring elements.

For a prime number $p$, a $p$-group is a group in which the order of each element is a power of $p$. A finite group is a $p$-group if and only if its order is a power of $p$.

The group of units of a ring is the set of its units, with the multiplication of the ring. The definition of the group of units gives the details.

A finitely generated group is a group that a finite subset generates. All finite groups and the group of integers are finitely generated.

An abelian group is a group whose operation is commutative. The integers with addition are the basic example.

The cyclic subgroup generated by an element of a group is the set of all integer powers of the element. It is the smallest subgroup that contains the element.

A group isomorphism is a bijective group homomorphism. Two isomorphic groups have the same structure, and differ only in the names of their elements.