What is the meaning of zero solution and non-zero solution in linear algebra? What does linear represent?

The zero solution in linear algebra refers to the linear secondary equation a. the X of equation AX = 0 can only take (0,0,0,0.). The explanation of non-zero is that except (0,0,0...) there are other vectors that can make it hold. I don't know what you mean by linear, which means linear algebra or linear equation
Ax = B. if B is 0, then it is a linear homogeneous equation. If B is not equal to 0, then it is a non-homogeneous linear equation

Given that the eigenvalues of matrix A = (A-41) (B 30) (0 02) are 1,1,2, find a, B; ask whether a is similar to diagonal matrix?
A=（a b 0）
-4 3 0
Ask for the guidance of the great God

A=
[a -4 1]
[b 3 0]
[0 0 2]
tr(A)=a+2+3=a+5=1+1+2=4
a=-1
det(A)=1*1*4=4
6a+8b=4
-6+8b=4
8b=10
b=5/4
Second, ask yourself to do it

From a α = λ α, P ^ - 1AP (P ^ - 1 α) = λ p ^ - 1 α is obtained,

Aα=λα
Multiply p ^ - 1 on both sides of the equation
P^-1Aα=λP^-1α
So p ^ - 1A (PP ^ - 1) α = λ p ^ - 1 α
So (P ^ - 1AP) (P ^ - 1 α) = λ p ^ - 1 α

Let a and B be real symmetric square matrices of order n, and a be positive definite, then there exists a real invertible matrix P such that p'ap = e, and p'bp = diag (λ 1 ,λn).

A real symmetric matrix must be similar diagonalized, positive definite, then all eigenvalues are greater than 0, so it is consistent with the identity matrix,

What is the meaning of zero solution and non-zero solution in linear algebra? What does linear represent?