a. Thenecessary and sufficient conditionsfor a non- empty subset W of V(F) to bea subspace are that :(i) a ≤ W, ße W⇒a-ße W(ii) a = F, a € W ⇒ aa € W.b. Obtain the smith normal form and rankof the matrix.A = [21 2 3450

(a)Suppose W is a subspace of a vector space V(F). Then if β∈W⇒-β∈W∴ α∈W, -β∈W⇒α+(-β)∈W (∵ W is…

Answered: a. The necessary and sufficient…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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6. Consider FV, where V consists of all real-valued functions. Let W C V, where W = {f € V | |f(x)| < M, Væ E R} i.e. W consists of all bounded functions. Show that W is a subspace of V. Let gV = M2(R), the vector space of all 2x2 matrices with real entries. Determine whether each of the following subsets of V is a subspace or not. (a) W = {A € M,(R)| A² = A} (1) w = {a € M6R) |a = [; ]} (b) W = {A € M2(R) |A = b
9. Let F be a field. Find bases of the following two subspaces of F5 : W₁ {(a1, a2, a3, a4, a5) € F5 | α₁ — a3 — a4 = 0} and a1 W₂ = {(α1, A2, A3, A4, A5 ) ≤ F5 | a2 = a3 = a₁ and a₁ + a5 = 0} What are the dimensions of W₁ and W₂ ? =
2. Let V = E3 and W = {x|x = a(1,0,2) + b(1,-1,3)} and S = {x|x = (x1,x2, 1)} where a and b are any real numbers. Is W a subspace of V or not?
Consider the following basis for the subspace of symmetric 2 × 2 real matrices B := For each of the following, find the associated symmetric 2 x 2 matrix A whose coordinate vector [A]g is the given column vector. 1. (1,2, 3)'. 2. (0, v2, 7)*. 3. (1,1, –1)'.
3. Let W be the set of all n x n diagonal matrices. Is W a subspace of the vector space Mn.n with the standard operations? Justify your answer.
a. Find a nonzero element X in W. X = Let V be the vector space of symmetric 2 x 2 matrices and W be the subspace 3][4] }. -4 -5 b. Find an element Y in V that is not in W. Y = W = span{ -1 -2 -2 -3
12. Prove that the set of m x n upper triangular matrices is a subspace of Mmxn (F).
Consider the following two subspaces of R4: U₁ = span[ 1 -3 2 ], and U₂ = span[ • What are the dimensions of U₁ and U₁? • What is the dimension of span [U₁, U₂]? 2 2 -2 0 0 What is the dimension of U₁n₂? Determine a basis for U₁U₂.
Let D = (d₁.d₂.d3) and F= {f₁.f2.13) be bases for a vector space V, and suppose f₁ = 6d, -d₂ +4d3, f₂=d₂ +d3, f3= -3d₁ +4d3. a. Find the change-of-coordinates matrix from F to D. b. Find [x]p for x=1₁-3f₂ +3f3. a. P = D+F
Define the subset W2 of M3 (R) to be the set of all matrices A € M3(R) satisfying a1,1 + a2,2 + a3,3 = 1. Explain why W2 is or is not a subspace of M3 (R).
4. Let H be the set of 2 x 2 whose trace is equal to zero. That is -{: a11 a12 Н — E M2(R) | a11 + a22 = A21 a22 Prove that H is a subspace of M2(R).
2. Prove the following given subsets of the vector space, M23, of all 2 x 3 matrices is a subspace. The set of all matrices of the form: [a b c] C], where b = a + c. [a b ] where a = -2c and f =2e + d. d e a. b.

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a. The necessary and sufficient conditions for a non- empty subset W of V(F) to be a subspace are that : (i) a ≤ W, ß ≤ W⇒a-ß¤W (ii) a ≤ F, a ≤ W ⇒ aa € W. 9. Obtain the smith normal form and rank of the matrix. A = [ 3] 123 450