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Linear Algebra

All vector spaces are finite dimensional

1) Let X be a finite dimensional linear space. Show that, for every subspace Y of X, Y perp perp = Y (that is, the annihilator of the annihilator of Y is Y)

2) X = U = V = polynomials in s, T=d/ds, S=multiplication by s
Show that S and T are linear and ST is not equal to TS

Solution Preview


In what follows, ^ stands for perp.

Since X is finite dimensional,

X = Y (+) Y^ (direct sum);

in particular, Y / Y^ = {0}

consequently, Y^ / Y^^ = ...