Spence Proof

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

Right now i really need this book for my class, but i dont have the $ right now. Is there anyone that has this book, and if so can you email me.

It's not the elementary Linear Algebra book, but the proof based 4th edition one.

Hmm I'll post the problems here. So don't close the question:

===================
Section 1.3

Definition. If S_1 and S_2 are nonempty subsets of a vector space V, then the sum of S_1 and S_2, denoted S_1 + S_2 is the set {x + y: x ∈ S_1 and y ∈ S_2}.

Definition. A vector space V is called the direct sum of W_1 and W_2 if W_1 and W_2 are subspaces of V such that W_1∩W_2 = {0} and W_1 + W_2 = V. We denote that V is the direct sum of W_1 and W_2 by writing V = W_1⊕ W_2.

(29) Let F be a field that is not of characteristic 2. Define

W_1 = {A ∈ M_(nxn) (F): A_(ij) = 0 whenever i ≤ j}

and W_2 to be the set of all symmetric n x n matrices with entries from F. Both W_1 and W_2 are subspaces of M_(nxn) (F). Prove that M_(nxn) (F) = W_1⊕ W_2. Compare this exercise with Exercise 28.

Here's problem (28):

A matrix M is called skew-symmetric if M^t = -M. Clearly, a skew-symmetric matrix is square. Let a F be a field. Prove that the set W_1 of all skew-symmetric n x n matrices with entries from F is a subspace of M_(nxn) (F). Now assume that F is not characteristic 2 (see Appendix C), and let W_2 be the subspace of M_(nxn) (F) consisting of all symmetric n x n matrices. Prove that M_(nxn) (F) = W_1⊕ W_2.

Here's the appendix part you'll need:

The smallest positive integer p for which a sum of p 1's equals 0 is called the characteristic of F; if no such positive integer exists, then F is said to have characteristic zero. Thus Z_2 has characteristic two, and R has characteristic zero.

(30) Let W_1 and W_2 be subspaces of a vector space V. Prove that V is the direct sum of W_1 and W_2 if and only if each vector in V can be uniquely written as x_1 + x_2, where x_1 ∈ W_1 and x_2 ∈ 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 S_1 and S_2 are arbitrary subsets of a vector space V, then span(S_1∪S_2) = span(S_1) + span(S_2). (The sum of two subsets is defined in the exercises of Section 1.3)

===================
Section 1.5

(18) Let S be a set of nonzero polynomials in P(F) such that no two have the same degree. Prove that S is linearly independent.

(20) Let f, g ∈ F(R, R) be the functions defined by f(t) = e^(rt) and g(t) = e^(st), where r ≠ s. Prove that f and g are linearly independent in F(R, R).

If you don't know what it means by F(R, R) here's the definition given in the text:

Let S be a nonempty set and W be any field and let F(S, W) denote 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 the operations of 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 familiar operations of addition and scalar multiplication of functions used in algebra and calculus.

===================
Section 1.6

(32) - (34) required the definitions I gave for Section 1.3

(32)

(a) Find an example of subspaces W_1 and W_2 of R³ with dimensions m and n, where m > n > 0, such that dim(W_1∩W_2) = n.

(b) Find an example of subspaces W_1 and W_2 of R³ with dimensions m and n, where m > n > 0, such that dim(W_1 + W_2) = m + n.

(c) Find an example of subspaces W_1 and W_2 of R³ with dimensions m and n, where m ≥ n, such that both dim(W_1∩W_2) < n and dim(W_1 + W_2) < m + n.

(33)

(a) Let W_1 and W_2 are subspaces of a vector space V such that V = W_1⊕W_2. If β_1 and β_2 are bases for W_1 and W_2, respectively, show that β_1∩β_2 ≠ ∅ and β_1∪β_2 is a basis for V.

(b) Conversely, let β_1 and β_2 be disjoint bases for subspaces W_1 and W_2, respectively, of a vector space V. Prove that if β_1∪β_2 is a basis for V, then V = W_1⊕W_2.

(34)

(a) Prove that if W_1 is any subspace of a finite-dimensional vector space V, then there exists a subspace W_2 of V such that V = W_1⊕W_2.

(b) Let V = R² 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 subspace of a finite-dimensional vector space V, and consider the basis {u_1, u_2, ..., u_k} for W. Let {u_1, u_2, ..., u_k, u_(k+1), ..., u_n} be an extension of this basis to a basis for V.

(a) Prove that {u_(k+1) + W, u_(k+2) + W, ..., u_n + W} is a basis for V/W.

(b) Derive a formula relating dim(V), dim(W), and dim(V/W).

The text says this for this question you should be familiar with question 31 from 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 the coset of W containing v. It if customary to denote this coset by v + W rather than

Bullet Proof (original)