1) Let V be a vector space, F a collection of subspaces of V with the following property: If X.YE F. then there exists a ZE F such that XUY C Z. Prove that UrerU is a subspace of V. Let V be a vector...

Question 6

Extracted text: 1) Let V be a vector space, F a collection of subspaces of V with the following property: If X.YE F. then there exists a ZE F such that XUY C Z. Prove that UrerU is a subspace of V. Let V be a vector space and assume that U, W are proper subspaces of V 2) and that U is not a subset of W and W is not a subset of U. Prove that UUWV is closed under scalar multiplication but is not a subspace of V. 3) Give an example of a vector space V and non-trivial subspaces X.Y, Z of V such that V = X@Y =X Z but Y Z. (Hint: You can find examples in R2.) 4) Let X, Y, Z be subspaces of a vector space V and assume that Y CX. Prove that Xn(Y+Z) = Y + (XnZ). This is known as the modular law of subspaces. 5) For each of the following subsets of F, determine whether it is a subspace of F: (where F is either C, R,or Q a) X2 X+ 2x2 + 3x3 = 0; %3D b) *2 + 2x2 + 3x3 = 4 c) X2 d) X2 5x3 %3D 6) Suppose b ER. Show that the set of continuous real-valued functions f on the intervai [0,1] such that , f(x) dx = b is a subspace of (C[0,1], R), the collection of all continuous functions (0,1] from to IR, if and only if b = 0. KSTAN 7) Suppose that U = EFx, y E F, , where F is either C, R,or Q. One can show that U is a subspace of F4you don't have to! Find a subspace V of F with F = U O V.