2. Below is a list V₂ of vector spaces over R. You are given subsets U; Vi. Decide which of these are subspaces. Justify your answers by giving a proof or a counter-example in each case. (1) V₁ = R¹ and U₁ = { (ao, a₁, A2, a3) € R4 | ²o a₁ = 0}. i=0 (2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c}. (3) V3 R³ and U3 = {(a + b, b, a+b): a, b ≤ R}. (4) V₁ = R³ and U₁ = {(a, b, c) € R³: a+b+c=0 and b + c =0}. (5) V5 = R³ and U5 = {(a, b, c) € R³: a = 0 or b = 0 or c = 0}. (6) V6 R[x] is the vector space of polynomials with real coefficients (this was example 3 in the lecture). U6 = {p E R[x] : p(1)=0}. (7) V7 = C([0, 1]) is the vector space of continuous functions from the interval [0, 1] to R. U+= {f e C([0, 1]) : fồf(z) - xdz =0}.
2. Below is a list V₂ of vector spaces over R. You are given subsets U; Vi. Decide which of these are subspaces. Justify your answers by giving a proof or a counter-example in each case. (1) V₁ = R¹ and U₁ = { (ao, a₁, A2, a3) € R4 | ²o a₁ = 0}. i=0 (2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c}. (3) V3 R³ and U3 = {(a + b, b, a+b): a, b ≤ R}. (4) V₁ = R³ and U₁ = {(a, b, c) € R³: a+b+c=0 and b + c =0}. (5) V5 = R³ and U5 = {(a, b, c) € R³: a = 0 or b = 0 or c = 0}. (6) V6 R[x] is the vector space of polynomials with real coefficients (this was example 3 in the lecture). U6 = {p E R[x] : p(1)=0}. (7) V7 = C([0, 1]) is the vector space of continuous functions from the interval [0, 1] to R. U+= {f e C([0, 1]) : fồf(z) - xdz =0}.
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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could you please help me with 5 and 6 and can you please provide explanations
![2. Below is a list V₂ of vector spaces over R. You are given subsets US Vi. Decide which
of these are subspaces. Justify your answers by giving a proof or a counter-example in each
case.
(1) V₁ = R¹ and U₁ = { (a0, a₁, A2, a3) € R¹ | Σ²_o a; = 0}.
i=0
(2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c}.
(3) V3 R³ and U3 =
{(a + b, b, a + b): a, b ≤ R}.
(4) V₁ = R³ and U₁ = {(a, b, c) € R³: a+b+c = 0 and b + c = 0}.
(5) V5 = R³ and U5 = {(a, b, c) € R³ : a =0 or b=0 or c = 0}.
(6) V6 R[x] is the vector space of polynomials with real coefficients (this was example
3 in the lecture). U6 = {p E R[x] : p(1)=0}.
(7) V7 = C([0, 1]) is the vector space of continuous functions from the interval [0, 1] to R.
U+= {f e C([0, 1]) : fồf(z) - xdz =0}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4061eb31-6ba2-4539-9b51-dd7b05481e7e%2F4b024e1d-508e-471b-84db-71d865eff3f1%2Fue0h3ag_processed.png&w=3840&q=75)
Transcribed Image Text:2. Below is a list V₂ of vector spaces over R. You are given subsets US Vi. Decide which
of these are subspaces. Justify your answers by giving a proof or a counter-example in each
case.
(1) V₁ = R¹ and U₁ = { (a0, a₁, A2, a3) € R¹ | Σ²_o a; = 0}.
i=0
(2) V₂ = R³ and U₂ = { (a, b, c) € R³ | ab = c}.
(3) V3 R³ and U3 =
{(a + b, b, a + b): a, b ≤ R}.
(4) V₁ = R³ and U₁ = {(a, b, c) € R³: a+b+c = 0 and b + c = 0}.
(5) V5 = R³ and U5 = {(a, b, c) € R³ : a =0 or b=0 or c = 0}.
(6) V6 R[x] is the vector space of polynomials with real coefficients (this was example
3 in the lecture). U6 = {p E R[x] : p(1)=0}.
(7) V7 = C([0, 1]) is the vector space of continuous functions from the interval [0, 1] to R.
U+= {f e C([0, 1]) : fồf(z) - xdz =0}.
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