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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s) The trace of a square n x n matrix A = (a) is the sum a₁ + a22+...+ aan of the entries on its main diagonal. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 0. Is H a subspace of the vector space V? 1. Is H nonempty? choose 12 5 6 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that H is not closed 3 4 7 under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.) " 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in IR and a matrix in H whose product is not in H 3 4 using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that is not closed 56 under scalar multiplication, it is…
1 10. Find the dimension of the subspace V = span{v1, v2, V3, v4} in R³ where vi = 1 2 1 V2 = 1 V3 = and v4 = 1 2 1 1
. (a) Suppose W₁ and W₂ are subspaces in a vector space V and that V = W₁ + W₂. Prove that V = W₁ W₂ if and only if any vector v EV can be represented uniquely as v = v₁ + v2 where V₁ € W₁, V2 € W2. (b) Suppose W₁ and W₂ are subspaces in a vector space V, and such that V = W₁ W₂. If B₁ is a basis of W₁ and 32 is a basis of W2 prove that B₁ U B2 is a basis of V.
Please answer with explanation. ASAP I will really upvote. Thanks
linear algebra 3.1 Q1 a, c, e
Solve for abstract algbera
Q.3. Determine if the set H of all matrices of the form [ is a subspace of M22. Here, p, q and r represent arbitrary real numbers. 04 Let H. be the set of all vectors of the form shown below Is H a subenace 2 Instify your
5. Let W₁ = { [au a11 a12 a22 laij € C} is a linear (vector) subspace of C2x2 with all o] laij € C} is a linear (vector) 0 { [an a21 a22 2 x 2 upper-triangular matrices and W₂ subspace of C2x2 with all 2 × 2 low-triangular matrices. Prove that W₁ W₂ is also a subspace of C2x2. And give a basis and the dimension of linear space W₁W₂. =
Determine which of the following sets are subspaces of R². 1. {[1, ₂] | 1 = -922}. 2. {[1, 2] | 6x₁ + 3x₂ = 0}. 3. {[1, ₂] | ₁ + x₂ > 0}. Determine which of the following sets are subspaces of R²x2. ? ? ? ? ? ? V 1. The 2 x 2 matrices with determinant equal to zero. ✓2. The 2 x 2 matrices whose entries are all greater than or equal to zero. ✓3. The 2 x 2 lower triangular matrices. Note: You can earn partial credit on this problem.

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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