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.

Reflexivity of equivalent matrices In linear algebra and matrix theory, there are two m×n matrices A and B. If these two matrices satisfy B=QAP(P is an n×n invertible matrix and Q is an m×m invertible matrix), then these two matrices are equivalent. That is to say, there is an invertible matrix, and A gets B through finite elementary transformation.

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Important theorem:

Every linear space has a base.

Yeah, one? n？ Okay？ n？ Non-zero matrix of column? A, if there is a matrix? b？ Manufacturing? AB？ =? Ba? =E(E is identity matrix), then? Answer? Is nonsingular matrix (or invertible matrix), and b is the inverse matrix of a.

A matrix is nonsingular (invertible) if and only if its determinant is not zero.

A matrix is nonsingular if and only if the linear transformation it represents is automorphism.

A matrix is semi-positive definite if and only if each eigenvalue is greater than or equal to zero.

A matrix is positive definite if and only if each eigenvalue is greater than zero.

Cramer's rule for solving linear equations.

Judge the relationship between augmented matrix and coefficient matrix of non-zero real root linear equations. ?

Baidu encyclopedia-linear algebra