= {x: x₁ = 2x₂} = {x: x₁ - x₂ = 2} = {x: x₁ = x₂ or x₁ = -x₂} = = {x: x₁ and x₂ are rational numbers} = {x: x₁ = 0} = {x: |x₁| + |x₂| = 0} 2. W = {x: x₁ = 2x3} 3. W = {x: x² = x₁ + x₂} 4. W = {x: x₂ = 0} 5. W = {x: x₁ = 2x3, x₂ = -x3} 6. W= {x: x3 = x2 = 2x1} : 7. W = {x: x₂ = x3 = 0} 8. Let a be a fixed vector in R³, and define W to be the subset of R³ given by W = {x: a¹x = 0}. Prove that W is a subspace of R³. Fuoroin 19 ho In each case, determine whether W is a subspace of R³. If W is a subspace, then give a geometric description of W. 9. W= {x: x3 = 2x1 - x₂} 10. W = {x: x₂ = x3 + x₁} 11. W= {x: X₁Xx₂ = X3} 3.2 Vector Space Properties of R" 26. In R4, let x = [1, -3, 2, 1], y = [2, 1, 3, 2], and z = [−3, 2, 1, 4]. Set a = 2 and b = -3. Illus- trate that the ten properties of Theorem 1 are satisfied by x, y, z, a, and b. 27. In R², suppose that scalar multiplication were de- fined by [x]-[2 X2 175 2ax1 2ax2 ax = a for every scalar a. Illustrate with specific examples those properties of Theorem 1 that are not satisfied. 28. Let
= {x: x₁ = 2x₂} = {x: x₁ - x₂ = 2} = {x: x₁ = x₂ or x₁ = -x₂} = = {x: x₁ and x₂ are rational numbers} = {x: x₁ = 0} = {x: |x₁| + |x₂| = 0} 2. W = {x: x₁ = 2x3} 3. W = {x: x² = x₁ + x₂} 4. W = {x: x₂ = 0} 5. W = {x: x₁ = 2x3, x₂ = -x3} 6. W= {x: x3 = x2 = 2x1} : 7. W = {x: x₂ = x3 = 0} 8. Let a be a fixed vector in R³, and define W to be the subset of R³ given by W = {x: a¹x = 0}. Prove that W is a subspace of R³. Fuoroin 19 ho In each case, determine whether W is a subspace of R³. If W is a subspace, then give a geometric description of W. 9. W= {x: x3 = 2x1 - x₂} 10. W = {x: x₂ = x3 + x₁} 11. W= {x: X₁Xx₂ = X3} 3.2 Vector Space Properties of R" 26. In R4, let x = [1, -3, 2, 1], y = [2, 1, 3, 2], and z = [−3, 2, 1, 4]. Set a = 2 and b = -3. Illus- trate that the ten properties of Theorem 1 are satisfied by x, y, z, a, and b. 27. In R², suppose that scalar multiplication were de- fined by [x]-[2 X2 175 2ax1 2ax2 ax = a for every scalar a. Illustrate with specific examples those properties of Theorem 1 that are not satisfied. 28. Let
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![1. W = {x: x₁ = 2x₂}
2. W = {x: x₁ - x₂ = 2}
3. W = {x: x₁= x2 or x₁ = -x₂}
4. W = {x: x₁ and x2 are rational numbers}
5. W = {x: x₁=0}
6. W {x: |x₁| + |x₂| = 0}
12. W = {x: x₁ = 2x3}
13. W = {x: x² = x₁ + x₂}
14. W = {x: x₂ = 0}
15. W = {x: x₁ = 2x3,
16. W = {x: x3 = x2 = 2x₁}
x₂ =−x3}
17. W = {x: x₂ = x3 = 0}
18. Let a be a fixed vector in R³, and define W to be the
subset of R³ given by
W = {x: a¹x = 0}.
Prove that W is a subspace of R³.
10 Lot Who the aubano.
Exorai
In each case, determine whether W is a subspace of R³.
If W is a subspace, then give a geometric description
of W.
9. W = {x: x3 = 2x₁ - xX₂}
10. W = {x: x2 = x3 + x₁}
11. W = {x: x₁x₂ = x3}
3.2 Vector Space Properties of R"
26. In R4, let x = [1, −3, 2, 1]ª, y = [2, 1, 3, 2], and
z = [-3, 2, 1, 4]. Set a = 2 and b = -3. Illus-
trate that the ten properties of Theorem 1 are satisfied
by x, y, z, a, and b.
175
27. In R2, suppose that scalar multiplication were de-
fined by
2ax1
-[x]-[2]
2ax2
X2
ax=a
for every scalar a. Illustrate with specific examples
those properties of Theorem 1 that are not satisfied.
28. Let](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1967bf9-9404-480e-a3d1-c9049b5b3e32%2F76f0a22c-1b3a-49e0-b729-e86c3057c9a8%2Fbjghxaa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. W = {x: x₁ = 2x₂}
2. W = {x: x₁ - x₂ = 2}
3. W = {x: x₁= x2 or x₁ = -x₂}
4. W = {x: x₁ and x2 are rational numbers}
5. W = {x: x₁=0}
6. W {x: |x₁| + |x₂| = 0}
12. W = {x: x₁ = 2x3}
13. W = {x: x² = x₁ + x₂}
14. W = {x: x₂ = 0}
15. W = {x: x₁ = 2x3,
16. W = {x: x3 = x2 = 2x₁}
x₂ =−x3}
17. W = {x: x₂ = x3 = 0}
18. Let a be a fixed vector in R³, and define W to be the
subset of R³ given by
W = {x: a¹x = 0}.
Prove that W is a subspace of R³.
10 Lot Who the aubano.
Exorai
In each case, determine whether W is a subspace of R³.
If W is a subspace, then give a geometric description
of W.
9. W = {x: x3 = 2x₁ - xX₂}
10. W = {x: x2 = x3 + x₁}
11. W = {x: x₁x₂ = x3}
3.2 Vector Space Properties of R"
26. In R4, let x = [1, −3, 2, 1]ª, y = [2, 1, 3, 2], and
z = [-3, 2, 1, 4]. Set a = 2 and b = -3. Illus-
trate that the ten properties of Theorem 1 are satisfied
by x, y, z, a, and b.
175
27. In R2, suppose that scalar multiplication were de-
fined by
2ax1
-[x]-[2]
2ax2
X2
ax=a
for every scalar a. Illustrate with specific examples
those properties of Theorem 1 that are not satisfied.
28. Let
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