Let a be an n-order square matrix, ξ be an n-dimensional sequence vector, a & # 178; ξ ≠ 0, a & # 179; ξ = 0, and prove that ξ, a ξ, a & # 178; ξ are linearly independent Linear algebra

Let a be an n-order square matrix, ξ be an n-dimensional sequence vector, a & # 178; ξ ≠ 0, a & # 179; ξ = 0, and prove that ξ, a ξ, a & # 178; ξ are linearly independent Linear algebra

N + 1 dimensional n-dimensional vector linear correlation, how to prove this?

A necessary and sufficient condition for linear correlation of vector groups α 1, α 2,..., α s is
The homogeneous linear equations (α 1, α 2,..., α s) x = 0 have nonzero solutions
For N + 1-dimensional n-dimensional vectors, because R (α 1, α 2,..., α n + 1)

N + 1 n-dimensional vector must be linearly related, and the linear phase is related to the linear independence, which is related to the solution of the system of equations, some of which I don't understand. In fact, the linear phase is related to the linear independence, which means whether there is a redundant equation, how can it be related to the solution?

Let's talk about the case of linear independence first. If n vectors are linearly independent, there are n useful equations (that is, the value of rank). At this time, 1. If the number of unknowns is greater than n (the number of unknowns is more than the number of equations), there must be infinite solutions; 2. If the number of unknowns is equal to n (n unknowns, n equations), then

Any n-dimensional vector can be represented by the n-dimensional vector group α 1. α 2 It is proved that α 1. α 2 α

It is proved that linearity has nothing to do with it!
It is proved that any n-dimensional vector can be divided into n-dimensional vector groups α 1, α 2 α n is linear
Therefore, the n-dimensional basic vector group ε 1, ε 2,..., ε n can be represented by α 1, α 2 α n is linear
And any n-dimensional vector can be expressed linearly by ε 1, ε 2,..., ε n
So the vector group ε 1, ε 2,..., ε N and α 1, α 2 α n is equivalent
So r (α 1, α 2 ,αn)=r(ε1,ε2,...,εn)=n.
So α 1, α 2 α n is linearly independent