clear explaination please Let U W be subspaces of V with di

clear explaination please :)

Let U W be subspaces of V with dim U = k and dim W = m, where k

19) Given that U is a subspace of W.

Since dim U = k < m (dim W) we can conclude that U is a proper subspace of W.

i.e. there is atleast one element in W which is not in U.

If say there is exactly one element x extra in W, then dim U along with x forms a basis.

Hence dim W = k+1

But since there is an l which lies between k and m, definitely m cannot be k+1

Or m > k+1

There should be atleast two elements x,y not in U but in W

Or there is a subspace with all elements in U with x alone which is a subspace of W.

Hence proved.