(a) The vector space P₂ has a basis {fi(r), f2(x), fa(x)} where fi(0) = f(0) = f(0) = 0. (b) Let U be a subspace of a vector space V. Suppose dim V = 4 and dim U = 2. Then every basis of V contains some basis of U as a subset. (e) Let U be a subspace of a vector space V. Suppose dim V = 6 and dim U = 2. There exists a 4-dimensional subspace W of V satisfying UCW CV. (d) Let S : V → W and T: UV be linear transformations. If ker S = {Ov} and ker T = {0} then ker (S o T) = {Ou}. (An equivalent statement is "If S and T are one-to-one then SoT is one-to-one".) (e) Let S : V→ W and T: UV be linear transformations. If ker (SoT) = {0} then ker S = {0} and ker T = {0}. (An equivalent statement is "If SoT is one-to-one then S and T are one-to-one".) (f) Let S : V→→W and T: UV be linear transformations. If im SW and im T = V then im So T = W. (An equivalent statement is "If S and T are onto then SoT is onto".) (g) Let S : VW and T: UV be linear transformations. If im So T = W then im S = W and im T = V. (An equivalent statement is "If SoT is onto then S and T are onto".) (h) Let T: UV be a linear transformation. If {v₁,..., Vk} is linearly independent in U then (T(v₁),...,T(va)} is linearly independent in V. (i) Let T: UV be a linear transformation. If {T(v₁),..., T(v)} is linearly independent in V then {V₁,..., Va} is linearly independent in U. (i) The linear transformation T: P₂ → R³ defined by T (p(x)) = [p(-1) p(0) P(1)] is invertible.
(a) The vector space P₂ has a basis {fi(r), f2(x), fa(x)} where fi(0) = f(0) = f(0) = 0. (b) Let U be a subspace of a vector space V. Suppose dim V = 4 and dim U = 2. Then every basis of V contains some basis of U as a subset. (e) Let U be a subspace of a vector space V. Suppose dim V = 6 and dim U = 2. There exists a 4-dimensional subspace W of V satisfying UCW CV. (d) Let S : V → W and T: UV be linear transformations. If ker S = {Ov} and ker T = {0} then ker (S o T) = {Ou}. (An equivalent statement is "If S and T are one-to-one then SoT is one-to-one".) (e) Let S : V→ W and T: UV be linear transformations. If ker (SoT) = {0} then ker S = {0} and ker T = {0}. (An equivalent statement is "If SoT is one-to-one then S and T are one-to-one".) (f) Let S : V→→W and T: UV be linear transformations. If im SW and im T = V then im So T = W. (An equivalent statement is "If S and T are onto then SoT is onto".) (g) Let S : VW and T: UV be linear transformations. If im So T = W then im S = W and im T = V. (An equivalent statement is "If SoT is onto then S and T are onto".) (h) Let T: UV be a linear transformation. If {v₁,..., Vk} is linearly independent in U then (T(v₁),...,T(va)} is linearly independent in V. (i) Let T: UV be a linear transformation. If {T(v₁),..., T(v)} is linearly independent in V then {V₁,..., Va} is linearly independent in U. (i) The linear transformation T: P₂ → R³ defined by T (p(x)) = [p(-1) p(0) P(1)] is invertible.
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
![Determine if each of the following statement is true or false.
(a) The vector space P₂ has a basis {fi(r), f2(r), fa(x)} where f1(0) = f2(0) = f(0) = 0.
(b) Let U be a subspace of a vector space V. Suppose dimV = 4 and dim U = 2. Then every basis of V
contains some basis of U as a subset.
(c) Let U be a subspace of a vector space V. Suppose dim V = 6 and dim U = 2. There exists a 4-dimensional
subspace W of V satisfying UCW CV.
(d) Let S : V → W and T: U → V be linear transformations.
If ker S = {Ov} and ker T = {0} then ker (SoT) = {Ou}.
(An equivalent statement is "If S and T are one-to-one then SoT is one-to-one".)
(e) Let S : V→ W and T: U → V be linear transformations.
If ker (So T) = {0} then ker S = {0} and ker T = {0}.
(An equivalent statement is "If SoT is one-to-one then S and T are one-to-one".)
(f) Let S : V → W and T: U → V be linear transformations.
If im S = W and im T = V then im So T=W.
(An equivalent statement is "If S and T are onto then SoT is onto".)
(g) Let S : V → W and T: U → V be linear transformations.
If im So T = W then im S = W and im T = V.
(An equivalent statement is "If S o T is onto then S and T are onto".)
(h) Let T: U → V be a linear transformation.
If {v₁,..., Vk} is linearly independent in U then {T(v₁),...,T(vk)} is linearly independent in V.
(i) Let T: U → V be a linear transformation.
If {T(v₁),...,T(vk)} is linearly independent in V then {v₁,..., Vk} is linearly independent in U.
(i) The linear transformation T: P₂ → R³ defined by T (p(x)) = [p(-1) p(0) p(1)] is invertible.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefe71c64-8346-40ed-b3ed-a6b5092127b6%2F76cfc0b8-269e-4b5c-a72c-acbf7c576899%2F1c7xqkk_processed.png&w=3840&q=75)
Transcribed Image Text:Determine if each of the following statement is true or false.
(a) The vector space P₂ has a basis {fi(r), f2(r), fa(x)} where f1(0) = f2(0) = f(0) = 0.
(b) Let U be a subspace of a vector space V. Suppose dimV = 4 and dim U = 2. Then every basis of V
contains some basis of U as a subset.
(c) Let U be a subspace of a vector space V. Suppose dim V = 6 and dim U = 2. There exists a 4-dimensional
subspace W of V satisfying UCW CV.
(d) Let S : V → W and T: U → V be linear transformations.
If ker S = {Ov} and ker T = {0} then ker (SoT) = {Ou}.
(An equivalent statement is "If S and T are one-to-one then SoT is one-to-one".)
(e) Let S : V→ W and T: U → V be linear transformations.
If ker (So T) = {0} then ker S = {0} and ker T = {0}.
(An equivalent statement is "If SoT is one-to-one then S and T are one-to-one".)
(f) Let S : V → W and T: U → V be linear transformations.
If im S = W and im T = V then im So T=W.
(An equivalent statement is "If S and T are onto then SoT is onto".)
(g) Let S : V → W and T: U → V be linear transformations.
If im So T = W then im S = W and im T = V.
(An equivalent statement is "If S o T is onto then S and T are onto".)
(h) Let T: U → V be a linear transformation.
If {v₁,..., Vk} is linearly independent in U then {T(v₁),...,T(vk)} is linearly independent in V.
(i) Let T: U → V be a linear transformation.
If {T(v₁),...,T(vk)} is linearly independent in V then {v₁,..., Vk} is linearly independent in U.
(i) The linear transformation T: P₂ → R³ defined by T (p(x)) = [p(-1) p(0) p(1)] is invertible.
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