In mathematics, a linear algebraic group is a subgroup of the group of invertible ${\displaystyle n\times n}$ matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation ${\displaystyle M^{T}M=I_{n}}$ where ${\displaystyle M^{T}}$ is the transpose of ${\displaystyle M}$.

Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.

One of the first uses for the theory was to define the Chevalley groups.

For a positive integer ${\displaystyle n}$, the general linear group ${\displaystyle GL(n)}$ over a field ${\displaystyle k}$, consisting of all invertible ${\displaystyle n\times n}$ matrices, is a linear algebraic group over ${\displaystyle k}$. It contains the subgroups

consisting of matrices of the form, resp.,

The group ${\displaystyle U}$ is an example of a unipotent linear algebraic group, the group ${\displaystyle B}$ is an example of a solvable algebraic group called the Borel subgroup of ${\displaystyle GL(n)}$. It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of ${\displaystyle \mathrm {GL} (n)}$ is conjugated into ${\displaystyle B}$. Any unipotent subgroup can be conjugated into ${\displaystyle U}$.

Another algebraic subgroup of ${\displaystyle \mathrm {GL} (n)}$ is the special linear group ${\displaystyle \mathrm {SL} (n)}$ of matrices with determinant 1.

The group ${\displaystyle \mathrm {GL} (1)}$ is called the multiplicative group, usually denoted by ${\displaystyle \mathbf {G} _{\mathrm {m} }}$. The group of ${\displaystyle k}$-points ${\displaystyle \mathbf {G} _{\mathrm {m} }(k)}$ is the multiplicative group ${\displaystyle k^{*}}$ of nonzero elements of the field ${\displaystyle k}$. The additive group ${\displaystyle \mathbf {G} _{\mathrm {a} }}$, whose ${\displaystyle k}$-points are isomorphic to the additive group of ${\displaystyle k}$, can also be expressed as a matrix group, for example as the subgroup ${\displaystyle U}$ in ${\displaystyle \mathrm {GL} (2)}$ :