# Tagged Questions

92 views

### linear recursion $y_n=A \cdot y_{n-1}$

Let $a,b, \in \mathbb{R}$. Let $x_0=a, x_1=b$ and $x_n=\frac{x_{n-1}+x_{n-2}}{2}$ for $n \geq 2$ (i) Write the recursion in the form $y_n=A \cdot y_{n-1}$ where $A$ is a $2 \times 2$ matrix and ...
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### Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
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chebyshev polynomials are defined as such: $T_n(x)=cos(n*arccos(x))$ I'm asked to show that $deg(T_j(x))=j$ and that $T_0,T_1,T_2,...,T_n$ are an orthogonal basis of $\mathbb R_n[x]$. I think I can ...
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### Recursive formulae involving a linear operator

Given a basis $e_{1}$, $e_{2}$ in the plane, define the linear operator $F$ as $F(e_{1})=3e_{1}+e_{2}$ and $F(e_{2})=e_{2}$. Furthermore, define the sequence $u_{1},u_{2},\dots$ of vectors in the ...
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### Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: $g(1) = \alpha;$ $g(2n+j) = 3g(n) + γn + β_j$ : j = 0,1 and n >= 1 I have assumed the closed form to ...
I came across the recursive sequence \begin{align} a_{n+1}&=(r-2)a_n+(r-1)b_n\;,\\ b_{n+1}&=a_n\;, \end{align} and the explicit formula $$a_n=(-1)^n(r-1)+(r-1)^n\;.$$ I saw that ...