Introduction To Linear Algebra(1)Linear Equation

SOLUTION SETS OF LINEAR SYSTEMS

齐次方程

齐次方程

Homogeneous Equation

The linear equations can be expressed in the form Ax=0, which is referred to as a homogeneous system. It inherently possesses the trivial solution x=0.

Nonhomogeneous equation

the solutions to the nonhomogeneous equation Ax = b constitute the set w = p + v_h, where p represents a particular solution to the equation Ax = b, and v_h denotes the homogeneous solution space corresponding to Ax = 0.

Linear Independence

The concept of linear independence is characterized by the equation x_1v_1 + x_2v_2 + \dots + x_pv_p = 0, which holds true exclusively when all coefficients x_i = 0. Consequently, the set \{v_1, v_2, \dots, v_p\} is termed a linearly dependent set.
A linear dependence relation among vectors v_1, \dots, v_p is defined by the equation c_1v_1 + c_2v_2 + \dots + c_pv_p = 0, where at least one coefficient c_i \neq 0.

Linear Transformation

The linear transformation T: \mathbb{R}^n \rightarrow \mathbb{R}^m can be uniquely represented by a matrix A . For any vector x \in \mathbb{R}^n , the transformation is given by multiplying the matrix A = [T(e_1) \dots T(e_n)] , where each column corresponds to the image of the standard basis vectors. The mapping is injective (one-to-one) if and only if for every vector b \in \mathbb{R}^m , there exists at most one vector x \in \mathbb{R}^n such that \( T(x) = b . The linear transformation T is injective precisely when the only solution to T(x) = 0 is the trivial vector x = 0.