In linear algebra, the column rank of matrix A is the largest number of linearly independent columns, which is usually expressed as r(A), rk(A) or rankA. In linear algebra, the column rank of matrix A is the maximum number of linearly independent columns of A. ..
Similarly, the row rank is the maximum number of linearly independent rows of A, that is, if the matrix is regarded as a row vector or a column vector, the rank is the rank of these row vectors or column vectors, that is, the number of vectors contained in the largest independent group.
1. Method of finding the rank of vector group: Construct the vector group matrix (a 1, ..., is) according to the column vectors. This matrix can also be transformed into trapezoidal matrix and non-zero rows, that is, the ranks of vector groups, by elementary row transformation.
2. Find the rank of the matrix: the matrix is transformed into a trapezoidal matrix by elementary row transformation, and the number of non-zero rows is the rank of the matrix.
3. The rank of the quadratic form is the rank of the matrix of the quadratic form: the rank is a linear algebraic term. In linear algebra, the rank of a matrix is the highest order number of its non-zero sub-formula, and the rank of a vector group is the number of vectors contained in its largest irrelevant group.
Linear algebra is a branch of mathematics.
Its research objects are vector, vector space (or linear space), linear transformation and finite dimensional linear equations. Vector space is an important subject in modern mathematics. Therefore, linear algebra is widely used in abstract algebra and functional analysis; Through analytic geometry, linear algebra can be expressed concretely.
The theory of linear algebra has been extended to operator theory. Because the nonlinear model in scientific research can usually be approximated as a linear model, linear algebra is widely used in natural science and social science.
Linear algebra is a branch of algebra, which mainly deals with linear relations.
Linear relationship means that the relationship between mathematical objects is expressed in linear form. For example, in analytic geometry, the equation of a straight line on the plane is a binary linear equation; The equation of spatial plane is a ternary linear equation, and the spatial straight line is regarded as the intersection of two planes, which is represented by an equation group composed of two ternary linear equations.
A linear equation with n unknowns is called a linear equation. Functions whose variables are linear are called linear functions. Linear relation problem is called linear problem for short. The problem of solving linear equations is the simplest linear problem. The so-called "linearity" refers to the following mathematical relationship.
Where f is called a linear operator or linear mapping. The so-called "algebra" refers to replacing elements and operations with symbols. In other words, we don't care whether X and Y above are real numbers or functions, nor whether F is polynomial or differential. We all abstract them into a symbol or a matrix.
Taken together, linear algebra studies what kind of linear operator F satisfies the linear relationship and what properties they have respectively.