Suppose F is a field with q elements and suppose V is a ndim

Suppose F is a field with q elements, and suppose V is a n-dimensional vector space over F. How many m-dimensional subspaces over F, V has?(m<n)

Solution

Its (n choose m) look at the basis of the vector space {v1, v2, v3 .. vn} choosing any m distinct vectors from here gives you a different subspace - no two can be the same because they will not contain the other basis vector.

Also all the subspaces can be represented like this - take any m dim vector space, look at all the vectors in the basis inside this subspace. If they are exactly m of them, they span the subspace and we are done, if they are less, then there is some vector they cannot represent. this vector can be written as a linear combination of the original v1 .. vn so at least one more of v1. .vn is inside the subspace.

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