Linear algebra : please solve q10 and 18 correctly and handwritten 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 thesubset of R³ given byW = {x: a¹x = 0}.Prove that W is a subspace of R³.10 Lot Who the aubano.ExoraiIn each case, determine whether W is a subspace of R³.If W is a subspace, then give a geometric descriptionof 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], andz = [-3, 2, 1, 4]. Set a = 2 and b = -3. Illus-trate that the ten properties of Theorem 1 are satisfiedby x, y, z, a, and b.17527. In R2, suppose that scalar multiplication were de-fined by2ax1-[x]-[2]2ax2X2ax=afor every scalar a. Illustrate with specific examplesthose properties of Theorem 1 that are not satisfied.28. Let

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Answered: = {x: x₁ = 2x₂} = {x: x₁ - x₂ = 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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Linear algebra : please solve q10 and 18 correctly and handwritten

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

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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