Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 1 / a

The minimal polynomial $m_T$ is the monic polynomial of least polynomial degree such that $m_T(T)=0$. Here a polynomial $p(x)={\sum }_j{c}_j{x}^j$ is evaluated at the linear operator by $p(T)={\sum }_j{c}_j{T}^j$, with the constant term acting as $c_{0}I$.

An annihilating polynomial exists because $I,T,T^2,\dots$ lie in the finite-dimensional vector space ${\mathrm{End}}_F(V)$ and are therefore eventually linearly dependent. Uniqueness follows by subtracting two monic annihilators of the same least degree: their difference has smaller degree and still annihilates $T$.

Equivalently,For the forward implication evaluate the factorization at $T$. For the reverse implication divide $p$ by $m_T$ using polynomial division; the remainder annihilates $T$, so minimality makes it zero. This also identifies the annihilating polynomials as the principal ideal generated by $m_T$.

Suppose $E^2=E$. Every vector has the decompositionIf $w=Ez$ also belongs to $\ker E$, then $w=E^{2}z=Ew=0$. HenceOn its image of a linear map, $E$ acts as the identity; on its kernel of a linear map, it acts as zero. Concatenate a basis of the image with a basis of the kernel. The resulting matrix is $\mathrm{diag}(I_r,0)$, where $r=\mathrm{rank}E$.

Conversely, this block matrix squares to itself, so any linear map having that matrix in a basis satisfies $E^2=E$. Thus projection operators are precisely the linear maps with this block form. This is an algebraic projection; no inner product or orthogonality is required.