grandes-ecoles 2025 Q14

grandes-ecoles · France · mines-ponts-maths2__psi Matrices Linear System and Inverse Existence
Show that $V A = B$ where $V \in \mathcal { M } _ { N } ( \mathbf { R } )$ is a matrix that you will make explicit.
Here $A = \left( a_1, a_2, \ldots, a_N \right)^\top \in \mathbf{R}^N$, $B = \left( \beta_0, \beta_1, \ldots, \beta_{N-1} \right)^\top \in \mathbf{R}^N$, and $\beta_k = \sum_{n=1}^{N} \lambda_n^k a_n$. 
Show that $V A = B$ where $V \in \mathcal { M } _ { N } ( \mathbf { R } )$ is a matrix that you will make explicit.

Here $A = \left( a_1, a_2, \ldots, a_N \right)^\top \in \mathbf{R}^N$, $B = \left( \beta_0, \beta_1, \ldots, \beta_{N-1} \right)^\top \in \mathbf{R}^N$, and $\beta_k = \sum_{n=1}^{N} \lambda_n^k a_n$.
Paper Questions
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