MAN-102, Tutorial Sheet-1: Linear Algebra - iitr.ac.in Spaces(1).pdf · MAN-102, Tutorial Sheet-1:...
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MAN-102, Tutorial Sheet-1: Linear Algebra (Vector Spaces and subspaces) Spring Semester- 2016-17 1. Let P 3 be the set of all real polynomials of degree ≤ 3 and let R be the field of real numbers. Define operations of addition ⊕ and the scalar multiplication ⊗ as follows: (a 0 + a 1 x + a 2 x 2 ) ⊕ (b 0 + b 1 x + b 2 x 2 )=(a 0 + b 0 )+(a 1 + b 1 )x +(a 2 + b 2 )x 2 α ⊗ (a 0 + a 1 x + a 2 x 2 )= αa 0 + αa 1 x + αa 2 x 2 Then show that (P 3 , ⊕, ⊗) is a real vector space. 2. Let R + be the set of all positive real numbers. Define operations of addition ⊕ and the scalar multiplication ⊗ as follows: u ⊕ v = uv for all u, v ∈ R + and α ⊗ u = u α for all u ∈ R + and α ∈ R (here R is the field of scalars). Prove that (R + , ⊕, ⊗) is a real vector space. 3. Show that the set of all n × n complex Hermitian matrices is a vector space over the field of real numbers under usual addition and scalar multiplication. 4. Let V be the set of all pairs (x, y) of real numbers, and let R be the field of real numbers. Define (x, y)+(x 1 ,x 2 )=(x + x 1 ,y + y 1 ) c(x, y)=(cx, y) Is V , with these operations, a vector space over the field of real numbers? 5. (a) Define an addition on R 2 by the rule (a 1 ,a 2 )+(b 1 ,b 2 )=(a 1 + b 1 , 0). Show that additive identity does not exist in R 2 w.r.t. above rule. (b) Define an addition on R 3 by the rule (a 1 ,a 2 ,a 3 )+(b 1 ,b 2 ,b 3 )=(a 1 b 1 ,a 2 b 2 ,a 3 b 3 ). Show that we have an additive identity for this operation in R 3 but inverse may not exist for some elements. 6. Let V = R 2 . Define operations of addition ⊕ and the scalar multiplication ⊗ as follows: (a 1 ,a 2 ) ⊕ (b 1 ,b 2 )=(a 1 + b 2 ,a 2 + b 1 ) and α ⊗ (a 1 ,a 2 )=(αa 1 , αa 2 ),α ∈ R (here R is the field of scalars). Does (V, ⊕, ⊗) form a real vector space? Give reasons for your assertion. 7. Let V = C n×n and F = C. Then show that the set of all Hermitian matrices is not a subspace of V if addition and scalar multiplication are defined in the usual way. 8. Let V = C[0, 1] be the set of all real valued function defined and continuous on the closed interval [0, 1]. Prove that V is a real vector space with respect to pointwise addition and multiplication. Further, determine that which of the following subsets of V are subspaces (a) {f ∈ V: f (1/2) = 0} (b) {f ∈ V: f (3/4) = 1} (c) {f ∈ V: f (0) = f (1)} (d) {f ∈ V: f (x) = 0 only at a finite number of points} 9. Determine whether each of the following set S form a subspace of R 4 , if addition and multiplication rules are defined in the usual way. (a) S = {(a, b, c, d): a = c + d}. (b) S = {(a, b, c, d): b = c - d and a = c + d}. (c) S = {(a, b, c, d): c = d}. 1
Transcript of MAN-102, Tutorial Sheet-1: Linear Algebra - iitr.ac.in Spaces(1).pdf · MAN-102, Tutorial Sheet-1:...
MAN-102, Tutorial Sheet-1: Linear Algebra(Vector Spaces and subspaces)
Spring Semester- 2016-17
1. Let P3 be the set of all real polynomials of degree ≤ 3 and let R be the field of real numbers.Define operations of addition ⊕ and the scalar multiplication ⊗ as follows:
(a0 + a1x+ a2x2)⊕ (b0 + b1x+ b2x
2) = (a0 + b0) + (a1 + b1)x+ (a2 + b2)x2
α⊗ (a0 + a1x+ a2x2) = αa0 + αa1x+ αa2x
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Then show that (P3,⊕,⊗) is a real vector space.
2. Let R+ be the set of all positive real numbers. Define operations of addition ⊕ and the scalarmultiplication ⊗ as follows: u⊕v = uv for all u, v ∈ R+ and α⊗u = uα for all u ∈ R+ and α ∈ R(here R is the field of scalars). Prove that (R+,⊕,⊗) is a real vector space.
3. Show that the set of all n× n complex Hermitian matrices is a vector space over the field of realnumbers under usual addition and scalar multiplication.
4. Let V be the set of all pairs (x, y) of real numbers, and let R be the field of real numbers. Define
(x, y) + (x1, x2) = (x+ x1, y + y1)
c(x, y) = (cx, y)
Is V , with these operations, a vector space over the field of real numbers?
5. (a) Define an addition on R2 by the rule (a1, a2) + (b1, b2) = (a1 + b1, 0). Show that additiveidentity does not exist in R2 w.r.t. above rule.
(b) Define an addition on R3 by the rule (a1, a2, a3) + (b1, b2, b3) = (a1b1, a2b2, a3b3). Show thatwe have an additive identity for this operation in R3 but inverse may not exist for someelements.
6. Let V = R2. Define operations of addition ⊕ and the scalar multiplication ⊗ as follows: (a1, a2)⊕(b1, b2) = (a1 + b2, a2 + b1) and α ⊗ (a1, a2) = (αa1, αa2), α ∈ R (here R is the field of scalars).Does (V,⊕,⊗) form a real vector space? Give reasons for your assertion.
7. Let V = Cn×n and F = C. Then show that the set of all Hermitian matrices is not a subspace ofV if addition and scalar multiplication are defined in the usual way.
8. Let V = C[0, 1] be the set of all real valued function defined and continuous on the closed interval[0, 1]. Prove that V is a real vector space with respect to pointwise addition and multiplication.Further, determine that which of the following subsets of V are subspaces
(a) {f ∈ V : f(1/2) = 0}(b) {f ∈ V : f(3/4) = 1}(c) {f ∈ V : f(0) = f(1)}(d) {f ∈ V : f(x) = 0 only at a finite number of points}
9. Determine whether each of the following set S form a subspace of R4, if addition and multiplicationrules are defined in the usual way.
(a) S = {(a, b, c, d) : a = c+ d}.(b) S = {(a, b, c, d) : b = c− d and a = c+ d}.(c) S = {(a, b, c, d) : c = d}.
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(d) S = {(−a+ c, a− b, b+ c, a+ b) : a, b, c ∈ R}.(e) S = {(a, b, c, d) : a = 1}.(f) S = {(a, b, c, d) : a ≤ b}.(g) S = {(a, b, c, d) : a = b = c = d}.(h) S = {(a, b, c, d) : a is an integer}.(i) S = {(a, b, c, d) : a2 − b2 = 0}.
10. Which of the following set of vectors α = (a1, a2, ..., an) in Rn are subspaces of Rn(n ≥ 3)?
(a) all α such that a1 ≥ 0;
(b) all α such that a1 + 3a2 = a3;
(c) all α such that a2 = a21;
(d) all α such that a1a2 = 0;
(e) all α such that a2 is rational.
11. Which of the following subsets of P are subspaces. Where, P is the real vector space of allpolynomials w.r.t. usual vector addition and scalar multiplication rules.
(a) {p ∈ P : deg. p ≤ 4}(b) {p ∈ P : deg. p = 4}(c) {p ∈ P : deg. p ≥ 4}(d) {p ∈ P : p(1) = 0}(e) {p ∈ P : p(2) = 1}(f) {p ∈ P : p′(1) = 0}
12. Which of the following subsets of M2×2 are subspaces. Note that, Mm×n is the vector space overreal field of all matrices of order m×n under usual definitions of addition and scalar multiplicationof matrices.
(a) All diagonal matrices.
(b) All upper triangular matrices.
(c) All symmetric matrices.
(d) All invertible matrices.
(e) All matrices which commute with a given matrix T .
(f) All matrices with zero determinant.
13. Let W1 and W2 be subspaces of a vector space V such that W1 ∪W2 is also a subspace. Showthat W1 ⊆ W2 or W2 ⊆ W1.
14. Let W1 and W2 be subspaces of a vector space V such that W1 +W2 = V and W1 ∩W2 = {0}.Show that for each vector u in V there are unique vectors u1 ∈ W1 and u2 ∈ W2 such thatu = u1 + u2.
Answers
4. No. 6. No. 8. (a) Yes (b) No (c) Yes (d) Yes9. (a) Yes (b) Yes (c) Yes (d) Yes (e) No (f) No (g) Yes (h) No (i) No10. (a) No (b) Yes (c) No (d) No11. (a) Yes (b) No (c) No (d) Yes (e) No (f) Yes12. (a) Yes (b) Yes (c) Yes (d) No (e) Yes (f) No
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