Spence Proof

9
Jun/090

Spence Proof
Linear Algebra by Arnold Insel, Spence, Lawrence, Stephen H.?

Now I really need this book to my class, but i dont have dollars in this time. Does anyone have this book, and if so you can send me an email. It is elementary linear algebra book, but the evidence for one based on the 4th edition.

Hmm I will post the problems here. So do not close the question: =================== Section 1.3 Definition. If S_1 and S_2 are subsets of a vector space V, then the sum of S_1 and s_2, S_1 + S_2 denotes the set (x + y: x ∈ S_1 and S_2 ∈ y). Definition. A vector space V is called the direct sum of w 1 and w 1 and W_2 if W_2 are subspaces of V such that w 1 ∩ W_2 = (0) and w 1 + W_2 = V. Denote that V is the direct sum of w 1 and V = w_1 writing W_2 ⊕ W_2 (29). Let F be a field that is 2 characteristics. Define w_1 = (A M_ ∈ (nxn) (F): A_ (IJ) = 0 when i ≤ j) and W_2 to be the set of all the nxn symmetric matrices with entries in F. Both w_1 and are subspaces W_2 M_ (nxn) (F). Show that M_ (nxn) (F) = w_1 ⊕ W_2. Compare this exercise with the exercise 28. Here is the problem (28): A matrix M is called antisymmetric if M ^ t =- M. Clearly, an antisymmetric matrix is square. Let a F a field. Show that the set of all skew-NX w_1 n symmetric matrices with entries from F is a subspace of M_ (nxn) (F). Now suppose F is not characteristic 2 (see Appendix C), and let the subspace of M_ W_2 (nxn) (F) consisting of all nxn symmetric matrices. Show that M_ (nxn) (F) = w_1 ⊕ W_2. Here is the part Appendix you need: P The smallest positive integer for which a sum of p 1 is equal to 0 is called the characteristic of F, if there is no positive integer, then F is said to have characteristic zero. So z_2 has features two, and R is zero property (30). Let w_1 and W_2 be subspaces of a vector space V. Show that V is the direct sum of w 1 and W_2 if and only if every vector in V can be written only x_1 + x_2, where x_2 and x_1 w_1 ∈ ∈ W_2. =================== Section 1.4 (12) Show that a subset of W of a vector space V is a subspace of V is, and only if (span W) = W. (14) Show that if the subsets S_1 and S_2 are arbitrary vector space V, then span (S_1 ∪ S_2) = span (S_1) + span (S_2). (We define the sum of the two subsets in the exercises section 1.3) =================== Section 1.5 (18) Let S be a set of nonzero polynomials P (F) such that no two have the same degree. Demonstrate S is linearly independent. (20) Let f, g ∈ F (R, R) is defined by the functions f (t) = e ^ (rt) and G (t) = e ^ (c), where r ≠ s. Demonstrate that f and g are linearly independent in F (R, R). If you do not know what it means for F (R, R), here is the definition given in the text: Let S be a set not empty and W be any field and let F (S, W) denotes the set of all functions from S to W. two functions f and g in F (S, W) are called equal if f (s) = g (s) for each s ∈ S. The set F (S, W) is a vector space with addition and scalar multiplication defined for f, g ∈ F (S, W) and c ∈ W by (f + g) (s) = f (s) + g (s) and (CF) (s) = c [f (s)] for each s ∈ S. Note that these are the families of the operations of addition and scalar multiplication of functions used in algebra and calculus. =================== Section 1.6 (32) - (34) requires the definitions given for the Section 1.3 (32) (a) Find an example of w_1 and W_2 subspaces of R ³ with dimensions m and n, where m> n> 0 such that dim (W_2 ∩ w_1) = n. (b) Find an example w_1 and W_2 of subspaces of R ³ with dimensions m and n, where m> n> 0 such that dim (w_1 + W_2) = m + n. (c) Find an example of w_1 and W_2 subspaces of R ³ with dimensions m and n, where n ≥ m, so that both dim (w_1 ∩ W_2) <n and dim (w_1 + W_2) <m + n. (33) (a w_1) Sea and W_2 are subspaces of a space vector V such that V = w_1 ⊕ W_2. If β_1 and β_2 are bases for w_1 and W_2, respectively, demonstrate that β_1 ∩ β_2 ≠ ∅ and β_1 ∪ β_2 is a basis for V. (b) Conversely, let β_1 and β_2 be the basis for w_1 and W_2 disjoint subspaces, respectively, of a vector space V. Show that if β_2 ∪ β_1 is a basis for V, V = w_1 ⊕ W_2. (34) (a) Show that if w 1 is any finite dimensional subspace of vector space V, then there exists a subspace of W_2 V such that V = w_1 ⊕ W_2. (B) Let V = R ² and w 1 = ((a_1, 0): a_1 ∈ R). Give examples of two different subspaces W_2 and W_2 'such that V = w_1 ⊕ W_2. and V = w_1 ⊕ W_2. (35) Let W be a finite dimensional subspace of vector space V, and consider the base (u_1, u_2, ..., W. u_k) (Let u_1, u_2, ... U_k, u_ (k +1), ..., U_n) be an extension of this basis, a basis for V. (a) Show that (u_ (k +1) + W, u_ (k + 2) + W, ..., U_n + W) is a basis for V / W. (b) Derive a formula that relates dim (V), dim (W) and dim (V / W). The text says that this question you should be familiar with the question 31 1.3 Here it is: Let W be a subspace of a vector space V over a field F. For any v ∈ V the set (v) + W = (v + w: w ∈ W) is called coset of W contains v. If used to denote this coset by V + W instead of