(9) (a) Let U, W be subspaces of V such that no proper subspace of V contains both U and W. If dimV = 7, dimU = 4 and dimW = 5, what is the dimension of V/(U ? W)? (b) Show that T ? L(V, W) is surjective, if and only if, W/ImT is trivial (i.e. it is equal to {0}). (c) Show that T ? L(V, W) is injective, if and only if, V/KerT is isomorphic to V by the quotient map.
Added by John R.
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Therefore, dim(V) = dim(U) + dim(W) - dim(U ∩ W). Given dim(V) = 7, dim(U) = 4, and dim(W) = 5, we can find the dimension of U ∩ W: 7 = 4 + 5 - dim(U ∩ W) dim(U ∩ W) = 2 Show more…
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