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Note: There is nothing special about three derivatives. This is j The same statement would be true for arbitrary n-th order linear ODsmple. ust an 8. Let V be a vector space. Let U and W be subspaces of V. (a) Prove: UnW is a subspace of v. (b) Prove: UuW is a subspace of V if and only if U C W or W CU (c) Let U+W:-: {u-น+10€ V: u E U and w e W}. Prove that U + W is a subspace of V (d) Prove: Suppose Unw-[0). Suppose that B is a finite basis for U and BI is a finite basis for W. Then BUBı is a finite basis for U+W. (It is then common to denote u +W as U#W, and call it the direct sum of U and W.) (e) Prove: Suppose U W is finite-dimensional. It is possible to choose a basis B for U and a basis Bi for W such that Bn B is a basis for U nw (f) Show by example that, in general, if B is a basis for U and B is a basis for W, then BnBi is NOT a basis for U n W

only part d,e,f

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