1. Let V be the set of all 2 x 2 matrices 4= such that the product abcd = 0 with standard operations on matrices a. Is V close under addition? Show your proof. b. Is V close under scalar multiplication? Show your proof. c. What is the zero vector in V? d. Does every matrix A in V have a negative that is in V? Explain e. Is V a vector space? Explain a 2. Is the set of vectors of the form b a subspace of R³? Show your proof. a+2b 3. Show that the vectors v₁ = (0, 3, 1, -1); v₂ (6, 0, 5, 1); v₁ =(4, -7, 1, 3) form a linearly dependent set in R¹? 4. Express V₁ in number 3 as linear combination of V2 and V3. 5. For which value of x do the following vectors form a linearly dependent set in R³ n=(x² = ²x=(2²+ x) =(²²x)
1. Let V be the set of all 2 x 2 matrices 4= such that the product abcd = 0 with standard operations on matrices a. Is V close under addition? Show your proof. b. Is V close under scalar multiplication? Show your proof. c. What is the zero vector in V? d. Does every matrix A in V have a negative that is in V? Explain e. Is V a vector space? Explain a 2. Is the set of vectors of the form b a subspace of R³? Show your proof. a+2b 3. Show that the vectors v₁ = (0, 3, 1, -1); v₂ (6, 0, 5, 1); v₁ =(4, -7, 1, 3) form a linearly dependent set in R¹? 4. Express V₁ in number 3 as linear combination of V2 and V3. 5. For which value of x do the following vectors form a linearly dependent set in R³ n=(x² = ²x=(2²+ x) =(²²x)
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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Question
ANSWER NO. 4
![1. Let V be the set of all 2 x 2 matrices 4=
4-[$]: such that the product abcd = 0 with standard operations on matrices
a. Is V close under addition? Show your proof.
b. Is V close under scalar multiplication? Show your proof.
c. What is the zero vector in V?
d. Does every matrix A in V have a negative that is in V? Explain
e. Is V a vector space? Explain
a
2. Is the set of vectors of the form
b a subspace of R³? Show your proof.
a+2b
3. Show that the vectors v₁ = (0, 3, 1, -1); v₂ = (6, 0, 5, 1); v₁ =(4, -7, 1, 3) form a linearly dependent set in R¹?
4. Express V₁ in number 3 as linear combination of V2 and V3.
5. For which value of x do the following vectors form a linearly dependent set in R³
*-[x = {m-G & G
x₂](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2ade466-87ff-48da-ae6a-268545b860cb%2F625315ea-c383-419c-8f8f-0e3011307588%2Fnkm6588_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let V be the set of all 2 x 2 matrices 4=
4-[$]: such that the product abcd = 0 with standard operations on matrices
a. Is V close under addition? Show your proof.
b. Is V close under scalar multiplication? Show your proof.
c. What is the zero vector in V?
d. Does every matrix A in V have a negative that is in V? Explain
e. Is V a vector space? Explain
a
2. Is the set of vectors of the form
b a subspace of R³? Show your proof.
a+2b
3. Show that the vectors v₁ = (0, 3, 1, -1); v₂ = (6, 0, 5, 1); v₁ =(4, -7, 1, 3) form a linearly dependent set in R¹?
4. Express V₁ in number 3 as linear combination of V2 and V3.
5. For which value of x do the following vectors form a linearly dependent set in R³
*-[x = {m-G & G
x₂
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