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Dept. Ch. of ECE, Faculty of Engineering University of Tehran
Linear Algebra Homework # I
Chapter 1: Vector Spaces Exercises to be handed: marked by *. Due date: Sunday Esfand 9th 1383. 1. Let V = { (a1 , a2) : a1 , a2 R }. Define addition of elements of V coordinate-wise, and for (a1 , a2) in V and c R, define:
(0 , 0) c(a1 , a 2 ) = (ca1 , a 2 c) if c = 0 if c 0
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Is V a vector space under these operations? Justify your answer. 2. Let V be the set of sequences {an} of real numbers. For any {an} and {bn} V and any real number , define {an} + {bn} = {an + bn} and .{an} = {.an}. Prove that V is a vector space under these operations. 3. Let V and W be vector spaces over a field F. Let Z = { (v , w) : v V and w W }. Prove that Z is a vector space over F under the operations (v1 , w1) + (v2 , w2) = (v1 + v2 , w1 + w2) and c.(v , w) = (c.v , c.w) .
4. How many elements are there in the vector space Mmxn(Z2) ? 5. Prove that the upper triangular matrices form a subspace of Mmxn(F). Notice: An mxn matrix A is called upper triangular if aij = 0 for i > j. 6. Let W1 and W2 be subspaces of a vector space V. Prove that W1 U W2 is a subspace of V if and only if W1 W2 or W2 W1. *7. Show that the set of convergent sequences {an} is a subspace of the vector space V in exercise 2.
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8. Let W1 denote the set of all polynomials f(x) in P(F) such that in the representation f(x) = an xn + an-1 xn-1 + + a0, we have aj = 0 whenever j is even. Likewise let W2 denote the set of all polynomials g(x) in P(F) such that in the representation g(x) = bn xn + bn-1 xn-1 + + b0, we have bj = 0 whenever j is odd. Prove that P(F) = W1 W2. 9. In Mmxn(F) define W1 denote the set of upper triangular matrices and W2 = { A Mmxn(F) : aij = 0 whenever i j}. Show that Mmxn(F) = W1 W2. *10. Let W be a subspace of a vector space V over a field F. For any v V let set {v} + W = { v + w : w W } is called the coset of W containing v. It is customary to denote v + W rather than {v} + W. Prove the following: a) v + W is a subspace of V if and only if v W. b) v1 + W = v2 + W if and only if v1 v2 W. *11. Let S denotes the collection (or class) of all cosets of W a subspace of vector space V over filed F. Addition and multiplication by elements of field F can be defined in S as follows: (v1 + W) + (v2 + W) = (v1 + v2) + W a.(v + W) = a.v + W
v1 and v2 V, v V and a F.
a) Prove that the operations above are well-defined; i.e., show that if v11 + W = v12 + W and v21 + W = v22 + W, then and (v11 + W) + (v21 + W) = (v12 + W) + (v22 + W) a.(v11 + W) = a.(v12 + W)
a F.
b) Prove that S is a vector space under the operations defined above. This vector space is called the quotient space of V modulo W is and denoted by V W .
12. Show that if S1 and S2 are arbitrary subsets of a vector space V, then:
L (S1 U S2) = L (S1) + L (S2).
Ch. 1 13. Let S1 and S2 be subsets of a vector space V. Prove that
L (S1 I S2) L (S1) I L (S2).
*14. Let S = { u1 , u2 , u3 , ., un } be a linearly independent subset of a vector space V over field Z2. How many elements are there in L (S)? Justify your answer. 15. Let V be a vector space over field F, and let u and v be distinct vectors in V. Prove that { u , v } is linearly independent if and only if { v + u and v u } are linearly independent. *16. Let M be a square upper triangular matrix with non-zero diagonal entries. Prove that columns of M are linearly independent. 17. Let S be a set of non-zero polynomials in P(F) such that no two elements of S has the same degree. Prove that S is linearly independent. 18. Prove that if { A1 , A2, . , Ak } is a linearly independent subset of Mnxn(F), then { A1T , A2T, . , AkT } is also linearly independent. 19. Let u and v be distinct elements of a vector space V. Show that if { u , v } is a basis for V and a and b are non-zero scalars, then both { u + v , a.u } and { a.u , b.v } are also bases for V. *20. Fins bases for the following subspaces of F5: W1 = { (a1 , a2 , a3 , a4 , a5) F5 : a1 a3 a4 = 0} and W2 = { (a1 , a2 , a3 , a4 , a5) F5 : a2 = a3 = a4 and a1 + a5 = 0 }. What are the dimensions of W1 and W2? 21. Prove that the set of all nxn matrices having trace equal to zero (denoted by W) is a subspace of Mnxn(F). Find a basis for W. What is the dimension of W?
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*22. Let V be a vector space having dimension n, and let S be a subset of V that generates V. a) Prove that there is a subset of S that is a basis for V (be careful not to assume that S is finite). b) Prove that S contains at least n elements. *23. Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x) Pn(R) there exist scalars c0 , c1 , , cn such that g(x) = c0f(x) + c1f(1)(x) + c2f(2)(x) + . + cnf(n)(x), where f(j)(x) denotes the jth derivatives of f(x). *24. Prove that if W1 is any subspace of a finite-dimensional vector space V, then there exists a subspace W2 of V such that V = W1 W2. *25. Let W be a subspace of a finite-dimensional vector space V, and consider the basis { u1 , u2 , , uk } for W. Let { u1 , u2 , , uk , uk+1 , uk+2 , , un } be an extension of this basis to a basis for V. Prove that { uk+1 + W , uk+2 + W, , un + W} is a basis for V W .
26. Let B be a subset of an infinite-dimensional vector space V. Then B is a basis for V if and only if for each non-zero vector v in V, there exist unique vectors u1 , u2 , , un in B and non-zero scalars c1 , c2 , , cn such that v = c1u1 + c2u1 + + cnun .