Homework Answers
1/2 b dr Problem 1: Suppose that [a, b] exists R, and let V be the space of all functions for which and is finite. For...
1/2 b dr Problem 1: Suppose that [a, b] exists R, and let V be the space of all functions for which and is finite. For two functions f and g in V and a scalar A e R, define addition and scalar multiplication the usual way: (Af)(x) f(x) f(x)g(r) and (fg)(x) Verify that the set V equipped with the above operations is a vector space. This space is called L2[a, b
1/2 b dr Problem 1: Suppose that [a,...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R and hom(V, W) be the set of all lnear transformations from the finite dimensional vector space V (dim V n and basis B) to the finite dimensional vector space W (dimW m and basis C) (i) Show with the usual addition and scalar multiplication of matrices, GLmRis a finite dimensional vector space, and dim GCmn(R) m Provide a basis B for (ii) Let VW...
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real...
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real valued functions on on the interval (-1, 1), with usual addition and scalar multiplication. (a) Verify, if the set W-f eV: f(0)-0is a subspace of V or not? (b) Verify, if the set W-Uev f(0) 1 is a subspace of V or not? (c) Verify, if the set W-İfEV:f(x)-0V-2-z is a subspace of V or not? 1b) PrtScn Home FS F6 F7 F8 5
Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the s...
Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the set S = {T : V -> F | T is a linear transformation}. Define the operations: (T1 + T2)(v) := T1(v) + T2(v), (aT1)(v) = a(T1(v)) for any v in V, a in F. Prove tat S with these operations is a vector space over F. (b) In S, we have elements fi : V -> F...
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...
1. Why the following sets are not vector space? with the regular vector addition and scalar...
1. Why the following sets are not vector space? with the regular vector addition and scalar multiplication. a) V = {E: * > 0, y 20 with the regula b) V = {l*: *y 2 o} with the regular vector addition and scalar multiplication. c) V = {]: x2+y's 1} with the regular vector addition and scalar multiplication. 2. The set B = {1,1+t, t + t2 is a basis for P, the set of all polynomials with degree less...
Need to use all axioms to prove this is a vector space. e(a+b)z and scalar multiplication...
Need to use all axioms to prove
this is a vector space.
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a vector space under these definitions? If so, what is the 0 element = eaeba- 8. Let V = k ea of V?
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of continuous functions from [0, 1 to R equipped with the usual function addition and scalar multiplication, is a vector space. (b) Let (V, K, +,-) be a vector space. Show that a non-empty subset W C V which is closed under and - necessarily contains the zero vector. (c) Is the set {(x,y)T: z,y E R, y a subspace of R2? Justify.
vectors pure and applied Exercise 11.3.1 Let Co(R) be the space of infinitely differentiable functions f R R. Show that CoCIR) is a vector space over R under pointwise addition and scalar multipl...
vectors pure and applied
Exercise 11.3.1 Let Co(R) be the space of infinitely differentiable functions f R R. Show that CoCIR) is a vector space over R under pointwise addition and scalar multiplication. Show that the following definitions give linear functionals for C(R). Here a E R. (i)8af f (a). minus sign is introduced for consistency with more advanced work on the topic of 'distributions'.) f(x) dx. (iii) J f-
Exercise 11.3.1 Let Co(R) be the space of infinitely differentiable...
Math 407 Homework 4 Name: 1. Why the following sets are not vector space? with the...
Math 407 Homework 4 Name: 1. Why the following sets are not vector space? with the regular vector addition and scalar multiplication. b) v = {(7: «y 20} with the regular vector addition and scalar multiplication. with the regular vector addition and scalar multiplication. 2. The set B = {1,1+t, t+t?} is a basis for P, the set of all polynomials with degree less than or equal to 2. Find the coordinate vector of p(t)-5+21+342 3. Let H =Span{ői, üz.us)...
1/2 b dr Problem 1: Suppose that [a, b] exists R, and let V be the space of all functions for which and is finite. For two functions f and g in V and a scalar A e R, define addition and scalar multiplication the usual way: (Af)(x) f(x) f(x)g(r) and (fg)(x) Verify that the set V equipped with the above operations is a vector space. This space is called L2[a, b
1/2 b dr Problem 1: Suppose that [a,...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R and hom(V, W) be the set of all lnear transformations from the finite dimensional vector space V (dim V n and basis B) to the finite dimensional vector space W (dimW m and basis C) (i) Show with the usual addition and scalar multiplication of matrices, GLmRis a finite dimensional vector space, and dim GCmn(R) m Provide a basis B for (ii) Let VW...
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real valued functions on on the interval (-1, 1), with usual addition and scalar multiplication. (a) Verify, if the set W-f eV: f(0)-0is a subspace of V or not? (b) Verify, if the set W-Uev f(0) 1 is a subspace of V or not? (c) Verify, if the set W-İfEV:f(x)-0V-2-z is a subspace of V or not? 1b) PrtScn Home FS F6 F7 F8 5
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...
1. Why the following sets are not vector space? with the regular vector addition and scalar multiplication. a) V = {E: * > 0, y 20 with the regula b) V = {l*: *y 2 o} with the regular vector addition and scalar multiplication. c) V = {]: x2+y's 1} with the regular vector addition and scalar multiplication. 2. The set B = {1,1+t, t + t2 is a basis for P, the set of all polynomials with degree less...
Need to use all axioms to prove
this is a vector space.
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a vector space under these definitions? If so, what is the 0 element = eaeba- 8. Let V = k ea of V?
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of continuous functions from [0, 1 to R equipped with the usual function addition and scalar multiplication, is a vector space. (b) Let (V, K, +,-) be a vector space. Show that a non-empty subset W C V which is closed under and - necessarily contains the zero vector. (c) Is the set {(x,y)T: z,y E R, y a subspace of R2? Justify.
vectors pure and applied
Exercise 11.3.1 Let Co(R) be the space of infinitely differentiable functions f R R. Show that CoCIR) is a vector space over R under pointwise addition and scalar multiplication. Show that the following definitions give linear functionals for C(R). Here a E R. (i)8af f (a). minus sign is introduced for consistency with more advanced work on the topic of 'distributions'.) f(x) dx. (iii) J f-
Exercise 11.3.1 Let Co(R) be the space of infinitely differentiable...
Math 407 Homework 4 Name: 1. Why the following sets are not vector space? with the regular vector addition and scalar multiplication. b) v = {(7: «y 20} with the regular vector addition and scalar multiplication. with the regular vector addition and scalar multiplication. 2. The set B = {1,1+t, t+t?} is a basis for P, the set of all polynomials with degree less than or equal to 2. Find the coordinate vector of p(t)-5+21+342 3. Let H =Span{ői, üz.us)...
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