Homework Answers
Since, in part (ii) only
addition and scalar multiplication of linear transformations is
defined but it is not mentioned what to do( like showing a vector
space or finding relationship between matrices and linear
transformations etc.). So I have answered this question assuming
that you need answer only for part(i).
Add Answer to:
(e) Let GLmn(R) be the set of all m x n matrices with entries in R...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
Let n EN Consider the set of n x n symmetric matrices over R with the...
Let n EN Consider the set of n x n symmetric matrices over R with the usual addition and multiplication by a scalar (1.1) Show that this set with the given operations is a vector subspace of Man (6) (12) What is the dimension of this vector subspace? (1.3) Find a basis for the vector space of 2 x 2 symmetric matrices (6) (16)
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 V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bi...
(e) Let V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all bilinear functions on V and W and the set GLn(m)R of all matrices of order n x m Hint: Uselal & rbl
(e) Let V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all...
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...
E finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R)...
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
Let V = M2x2 be the vector space of 2 x 2 matrices with real number...
Let V = M2x2 be the vector space of 2 x 2 matrices with real number entries, usual addition and scalar multiplication. Which of the following subsets form a subspace of V? The subset of upper triangular matrices. The subset of all matrices 0b The subset of invertible matrices. The subset of symmetric matrices. Question 6 The set S = {V1, V2,v;} where vi = (-1,1,1), v2 = (1,-1,1), V3 = (1,1,-1) is a basis for R3. The vector w...
Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and...
Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and scalar multiplication, and Wー State the incorrect statement from the following five 1. W is a subspace of GL2(R) with basis 2. W -Ker f, where GL2(R) R is the linear transformation defined by: 3. Given the basis B in option1. coordB( 23(1,2,2) 4. GC2(R)-W + V, where: 5. Given the basis B in option1. coordB( 2 3 (1,2,3) Problem 5....
Problem 3. Let D be the vector space of all differentiable function R wth the usual pointwise add...
Problem 3. Let D be the vector space of all differentiable function R wth the usual pointwise addition and scalar multiplication of functions. In other words, for f, g E D and λ E R the function R defined by: (f +Ag) ()-f(r) +Ag(x) Let R be four functions defined by: s(x)-: sin 11 c(r) : cosz, co(z)--cos(z + θ), and so(r) sin(z + θ), and Wspanls, c Which of the following statements are true: (a) For each fixed θ...
4. Bonus 8 pt total] This question is about dual vector spaces and their connection to the transp...
4. Bonus 8 pt total] This question is about dual vector spaces and their connection to the transpose If V is a vector space, then the vector space Hom(V, R), of linear maps from V to R is called the dual vector space of V and we denote it by V". a) Suppose ei Is a basis for V. Show or which we have I i=j (b) Show that the set e is linearly independent. (c) If V is finite...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
Let n EN Consider the set of n x n symmetric matrices over R with the usual addition and multiplication by a scalar (1.1) Show that this set with the given operations is a vector subspace of Man (6) (12) What is the dimension of this vector subspace? (1.3) Find a basis for the vector space of 2 x 2 symmetric matrices (6) (16)
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 V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all bilinear functions on V and W and the set GLn(m)R of all matrices of order n x m Hint: Uselal & rbl
(e) Let V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all...
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...
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
e finite dimensional vector Spaces, dim Show that there is a bijection between the set Bil(V,W;R) of all bilinear functions on V and W and the set ĢLn(m)R of all matrices of order n x m.
Let V = M2x2 be the vector space of 2 x 2 matrices with real number entries, usual addition and scalar multiplication. Which of the following subsets form a subspace of V? The subset of upper triangular matrices. The subset of all matrices 0b The subset of invertible matrices. The subset of symmetric matrices. Question 6 The set S = {V1, V2,v;} where vi = (-1,1,1), v2 = (1,-1,1), V3 = (1,1,-1) is a basis for R3. The vector w...
Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and scalar multiplication, and Wー State the incorrect statement from the following five 1. W is a subspace of GL2(R) with basis 2. W -Ker f, where GL2(R) R is the linear transformation defined by: 3. Given the basis B in option1. coordB( 23(1,2,2) 4. GC2(R)-W + V, where: 5. Given the basis B in option1. coordB( 2 3 (1,2,3) Problem 5....
Problem 3. Let D be the vector space of all differentiable function R wth the usual pointwise addition and scalar multiplication of functions. In other words, for f, g E D and λ E R the function R defined by: (f +Ag) ()-f(r) +Ag(x) Let R be four functions defined by: s(x)-: sin 11 c(r) : cosz, co(z)--cos(z + θ), and so(r) sin(z + θ), and Wspanls, c Which of the following statements are true: (a) For each fixed θ...
4. Bonus 8 pt total] This question is about dual vector spaces and their connection to the transpose If V is a vector space, then the vector space Hom(V, R), of linear maps from V to R is called the dual vector space of V and we denote it by V". a) Suppose ei Is a basis for V. Show or which we have I i=j (b) Show that the set e is linearly independent. (c) If V is finite...
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