Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that
\[XY-YX=I.\]
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Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

(a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

(b) Show that matrices
\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.