Problem 1: Prove or disprove each of the following statements. (1) The union of two subspaces of a vector space is a subspace. (2) S = {(x, y, z) = R³: xyz = 0} is a subspace of R³. (3) S = {(x, y, z, w) ER¹ : x-y + 2z = 0 and w = 1} is a subspace of R¹. (4) In R³ the vectors (1, 1, 1) and (-1, t, 1) are linearly independent for any scalar t. (5) The set S = {1,1+x} is a basis for P₁. (6) The set S= {1, x, 1+x} spans P₁. (7) The set S= {3+x,3+2x-x², 2³} spans P3. (8) The set S= {1, 2, 1-x²} is a linearly independent subset of P2. (9) The set S = {1, 2, 1-2} is a basis for P2. [111] 22 2 (10) The nullity of the matrix is 2.
Problem 1: Prove or disprove each of the following statements. (1) The union of two subspaces of a vector space is a subspace. (2) S = {(x, y, z) = R³: xyz = 0} is a subspace of R³. (3) S = {(x, y, z, w) ER¹ : x-y + 2z = 0 and w = 1} is a subspace of R¹. (4) In R³ the vectors (1, 1, 1) and (-1, t, 1) are linearly independent for any scalar t. (5) The set S = {1,1+x} is a basis for P₁. (6) The set S= {1, x, 1+x} spans P₁. (7) The set S= {3+x,3+2x-x², 2³} spans P3. (8) The set S= {1, 2, 1-x²} is a linearly independent subset of P2. (9) The set S = {1, 2, 1-2} is a basis for P2. [111] 22 2 (10) The nullity of the matrix is 2.
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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![Problem 1: Prove or disprove each of the following statements.
(1) The union of two subspaces of a vector space is a subspace.
(2) S = {(x, y, z) = R³: xyz = 0} is a subspace of R³.
(3) S = {(x, y, z, w) = R¹ : x-y + 2z = 0 and w=1} is a subspace of R¹.
(4) In R³ the vectors (1, 1, 1) and (-1, t, 1) are linearly independent for any
scalar t.
(5) The set S = {1,1+x} is a basis for P₁.
(6) The set S =
{1, x, 1+x} spans P₁.
(7) The set S = {3+x, 3+2x-x², 2³} spans P3.
(8) The set S= {1, x, 1-x²} is a linearly independent subset of P2.
(9) The set S= {1, x, 1-x²} is a basis for P2.
[1 1 1]
22 2
(10) The nullity of the matrix
is 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe469c8e1-3d66-4aef-8026-c4bcbe49028e%2F688dccd6-c808-4628-85b9-cc5238d4e7b8%2Ft3xrj7k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1: Prove or disprove each of the following statements.
(1) The union of two subspaces of a vector space is a subspace.
(2) S = {(x, y, z) = R³: xyz = 0} is a subspace of R³.
(3) S = {(x, y, z, w) = R¹ : x-y + 2z = 0 and w=1} is a subspace of R¹.
(4) In R³ the vectors (1, 1, 1) and (-1, t, 1) are linearly independent for any
scalar t.
(5) The set S = {1,1+x} is a basis for P₁.
(6) The set S =
{1, x, 1+x} spans P₁.
(7) The set S = {3+x, 3+2x-x², 2³} spans P3.
(8) The set S= {1, x, 1-x²} is a linearly independent subset of P2.
(9) The set S= {1, x, 1-x²} is a basis for P2.
[1 1 1]
22 2
(10) The nullity of the matrix
is 2.
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