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14. Let V be the set of two elements given by fTRUE, FALSE. Define the addition as TRUE if eractl...
2. Let S be the set of all functions from R to R. For f.g es,...
2. Let S be the set of all functions from R to R. For f.g es, we define the binary operation on S by (fog)(x) = f(x) + g(x) + 3x*, VX E R. (1) Find the additive identity in S under the operation . (ii) Find the additive inverse of the function w es defined by w(x) = 5x - 8, VXER [4] under the operation .
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define...
Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) @ (x, y) = (u + x +1, v + y + 1) for all (u, v) and (x, y) in V. Define an operation of "scalar multipli- cation" by a® (x, y) = (ax, ay) for all a E R and (x,y) E V Under the two operations the set V is not a vector space....
Question 1 (10 Marks) This question consists of 10 true false ansers. In cach ease, answer true i...
Question 1 (10 Marks) This question consists of 10 true false ansers. In cach ease, answer true if the statement is always true and false otherise. If a statement is false, 1. The set rER0 isa group under the binary operation o defined ad-be is a group under matrix addition. 3. Tho sot eRzs not an Abelian group under the binary erplain why. There is no need to show working for true statements. by a ob vab. 2. The set...
Let V be the set of all 3x3 matrices with Real number entries, with the usual...
Let V be the set of all 3x3 matrices with Real number entries, with the usual definitions of scalar multiplication and vector addition. Consider whether V is a vector space over C. Mark all true statements (there may be more than one). e. The additive inverse axiom is satisfied f. The additive closure axiom is not satisfied g. The additive inverse axiom is not satisfied h. V is not a vector space over C i. The additive closure axiom is...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
(a) State whether the following statement is true or false. The follow set is a subspace...
(a) State whether the following statement is true or false. The follow set is a subspace of P2, where P2 is the set of all polynomials over the real numbers of degree 2 or less. W={p € P2 :p (3)=0} O True O Fale In the essay box below, if it is true, prove that W is closed under scalar multiplication. Otherwise, give an explantion why the statement is false. XDX HE Editor A-AIBIU S *** Styles Font Size Words:...
QUESTION 3 Let V be the set of column vectors with two Complex number entries with...
QUESTION 3 Let V be the set of column vectors with two Complex number entries with the following definitions of vector addition and scalar multiplication, X + 2 w - 22 []+[%]-[*+27] Is Va vector space over the field of Complex numbers? Why or why not? a. Yes, because all 10 vector space axioms are satisfied b. No, because neither the Zero axiom nor the Additive inverse axiom is satisfied O No, because though the Additive Inverse axiom is satisfied,...
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an inner product on Rmxn. (d) Consider a periodic-function space V with period of 1 sec. Define an inner product on V by <f,a>= f(t )a (t ) dt. Then cos 2 π t...
2. Let S be the set of all functions from R to R. For f.g es, we define the binary operation on S by (fog)(x) = f(x) + g(x) + 3x*, VX E R. (1) Find the additive identity in S under the operation . (ii) Find the additive inverse of the function w es defined by w(x) = 5x - 8, VXER [4] under the operation .
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) @ (x, y) = (u + x +1, v + y + 1) for all (u, v) and (x, y) in V. Define an operation of "scalar multipli- cation" by a® (x, y) = (ax, ay) for all a E R and (x,y) E V Under the two operations the set V is not a vector space....
Question 1 (10 Marks) This question consists of 10 true false ansers. In cach ease, answer true if the statement is always true and false otherise. If a statement is false, 1. The set rER0 isa group under the binary operation o defined ad-be is a group under matrix addition. 3. Tho sot eRzs not an Abelian group under the binary erplain why. There is no need to show working for true statements. by a ob vab. 2. The set...
Let V be the set of all 3x3 matrices with Real number entries, with the usual definitions of scalar multiplication and vector addition. Consider whether V is a vector space over C. Mark all true statements (there may be more than one). e. The additive inverse axiom is satisfied f. The additive closure axiom is not satisfied g. The additive inverse axiom is not satisfied h. V is not a vector space over C i. The additive closure axiom is...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
(a) State whether the following statement is true or false. The follow set is a subspace of P2, where P2 is the set of all polynomials over the real numbers of degree 2 or less. W={p € P2 :p (3)=0} O True O Fale In the essay box below, if it is true, prove that W is closed under scalar multiplication. Otherwise, give an explantion why the statement is false. XDX HE Editor A-AIBIU S *** Styles Font Size Words:...
QUESTION 3 Let V be the set of column vectors with two Complex number entries with the following definitions of vector addition and scalar multiplication, X + 2 w - 22 []+[%]-[*+27] Is Va vector space over the field of Complex numbers? Why or why not? a. Yes, because all 10 vector space axioms are satisfied b. No, because neither the Zero axiom nor the Additive inverse axiom is satisfied O No, because though the Additive Inverse axiom is satisfied,...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an inner product on Rmxn. (d) Consider a periodic-function space V with period of 1 sec. Define an inner product on V by <f,a>= f(t )a (t ) dt. Then cos 2 π t...
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