Let A be a finite dimensional vector space of dimensionm over field F. Find a na

Question

Let A be a finite dimensional vector space of dimensionm over field F. Find a natural bijection (doesn't depend on choice ofbasis) between the following sets 1) subspaces of V 2) subspaces of V* (the dual space of V) 3) quotient spaces of V 4) quotient spaces of V* Let A be a finite dimensional vector space of dimensionm over field F. Find a natural bijection (doesn't depend on choice ofbasis) between the following sets 1) subspaces of V 2) subspaces of V* (the dual space of V) 3) quotient spaces of V 4) quotient spaces of V*

Explanation / Answer

the set of all linear transformations those can be formed froma subspace V to itself also forms a vector space denoted byL(V,V) and called the dual space of V = V* in the beginning , you wrote the vector space to be Aand later referring it as V. so, i am considering the vector space to be V. we know that V is the subspace of itself. also, the null space {0} is a subspace of V. so, the quotient space is V/ {0} is isomorphic toV.   remember the dimension theorem dim V - dim W = dimV/W where W is a subspace of V. so, V/{0} --> V is the natural bijection . similarly, {0^} the Zero transformation is the trivialsubspace of V* so, V* / {0^} is a bijection to V*.

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