second time puzzled.

Heres an interesting linear algebra problem: Let F be a field and let V be a finite dimensional vector space over F. If α₁ , ... , α_m are finitely many vectors in V, each different from zero vector, prove that there is a linear functional f on V such that


     f(α_i) ≠ 0 , i = 1 , .., m

My attempts at proving this lead to this problem: Can a union of finitely many sub spaces of dimension N-1 (hyperplanes) cover a vector space of dimension N.? Which seems an interesting problem in itself.

Not long ago, I happened to visit Pune again for a day to settle an ancient but pressing matter. A tight schedule notwithstanding, I could squeeze in some time to visit one of my old haunts - Popular book house and walked out with a book which caught my attention-"i have landed" by s.j. gould( blissfully ignoring its pricey tag).

In the afternoon, back in the train, as people were settling down in their seats I began reading an essay from the book "no science without fantasy , no art without facts", which using Nabokov's example argues that a ``restless scholar'' whose work straddles different disciplines of knowledge can indeed enrich the fields and not become just a jack of all trades and master of none. Nabokov, we all know was literary giant of his times, but his interests in lepidoptery and his career as a "lab drudge" are less well known. Quoting nabokov, Gould writes----

...I work on my personal research .. a study of the classification of american 'blues' based on he structure of their genitalia(miniscule sculpteresque hooks, teeth, spurs etc visible only under microscope), while I sketch in with the aid of various marvelous devices, variants of the magic lantern..My work utterly exhausts me ... To immerse yourself in the wondrous crystralline world of the microscope, where silence reigns, circumscribed by its own horizon, a blindingly white arena--all this is so enticing that I cannot describe it.

Nabokov obsession with detail is dealt with at great length in the essay and Gould's unfettered admiration and bubbling enthusiasm is all too apparent beneath his own precisely crafted sentences though he tries to subdue it somewhat with pedantic words. I must say I got infected too.

Again, quoting Nabokov he writes-- Both the writer of fiction and the naturalist drew on a profound delight in precise comparitive observation. For Nabokov, a work of nature was a like work of art or rather it was a profound work of art, by the greatest of all living artists, evolution, and as much a joy to the mind and a challenge to the intellect as a Shakespear's sonnet. Hence it deserved to be studied like it, with never ending attention to detail and patience.

As I wonder about all this, an impulse prompted me to look outside the train. It had not started yet. A rat manoevered through the turd in the tracks. Somebody throws out a unused plastic bag with Sambar like liquid in it down the tracks. A scavenger notices this and rushes to it and grabs it before the a rat has it and starts sucking the Sambar from the plastic. The train begins to move, or rather wallow through the turd of the tracks. I used to think a train journey can bring the mind to sharp focus but this left me completely befuddled. The barrage of senses -- smells, sights and thoughts was overwhelming. As I was about to bid my Nabokovian delights goodbye, the greenery of the landscape engulfs my view. A person occupies a seat in front as for time and engages me in chit chat. But I never recovered.