Let p be an orthogonal matrix of order n and X be a column vector of n-dimensional unit length, then |||||= ()? What do two verticals mean?

Let p be an orthogonal matrix of order n and X be a column vector of n-dimensional unit length, then |||||= ()? What do two verticals mean?

||PX | is the length of PX
Since P is an orthogonal matrix, so |||||||= |||||||||= 1

What is the n-dimensional column vector in linear algebra? What exactly does it look like, one row of N columns or n rows of N columns, or n rows of N columns?

An n-dimensional column vector is n rows and 1 column
The n-dimensional row vector is one row and n-column
Intuitively
The column vector is one column
The row vector is one row

Now that we know that a is an n-dimensional unit and l-column vector, how do we deduce "so" in the following steps?
"Given that the eigenvalues of AA ^ t are 1,0,0,..., 0, the eigenvalues of a = e-AA ^ t are 0,1,1,..., 1"

Because the eigenvalues of AA ^ t are 1,0,0,0, which are determined by | λ e-AA ^ t | = 0, there are | e-AA ^ t | = 0 and | AA ^ t | = 0. Then we see | λ e-A | = | λ e - (e-AA ^ t) | = | (λ - 1) e + AA ^ t | = | AA ^ t - (1 - λ) e | when λ = 1, it is the same as the 0 of the original characteristic equation of AA ^ T. when λ = 0, it is the same as the 1 of the original characteristic equation of AA ^ t

Let α 1, α 2 α n is a set of n-dimensional vectors. The n-dimensional unit coordinate vectors E1, E2 It is proved that α 1, α 2 α n is linearly independent

It is proved that any group of n-dimensional vectors can be expressed linearly by the group of n-dimensional unit vectors, that is, α 1, α 2 α n can be represented by n-dimensional unit coordinate vectors E1, E2 The n-dimensional unit coordinate vectors E1, E2 The results show that, en can be determined by α 1, α 2 α n is a linear representation of "∩ α 1, α 2 α n is the unit coordinate vector E1, E2 , en equivalent ∧ R (α 1, α 2,...) ，αn）=r（e1，e2，… ，en）=n∴α1，α2，… α n is linearly independent