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3. (10 points) Let F denote the vector space of functions f: R + R over...
(3) Let V denote a vector space over the field F and let v,..., Un E...
(3) Let V denote a vector space over the field F and let v,..., Un E V. (a) Show that span(vn, 2,. , Un) (b) Show that span (ui , U2 , . . . , vn) span(v)+ +span(vn). span(v1)@span(v2)㊥·..㊥8pan(vn) if and only if (vi , . , . , %) is linearly independent.
Problem 6. Let Coo(R) denote the vector space of functions f : R → R such...
Problem 6. Let Coo(R) denote the vector space of functions f : R → R such that f is infinitely differentiable. Define a function T: C (RCo0 (R) by Tf-f -f" a) Prove that T is a linear map b) Find a two-dimensional subspace of null(T).
2. (a) Let B = {f1, f2, f3} be a subset of P2 where fi(x) =...
2. (a) Let B = {f1, f2, f3} be a subset of P2 where fi(x) = x² – 3, f2(x) = x2 – 2x and f3(x) = x. Show that B is a basis of P2. (b) Determine whether or not the following sets are subspaces of F. (i) X = {f € F | f(x) = a(x + cos x), a € R}. (ii) Y = {f EF | f(x) = ax + sin x, a € R}. (c)...
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...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ]...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
Problem 9. Let V be a vector space over a field F (a) The empty set...
Problem 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...
6. (a) Let V be a vector space over the scalars F, and let B =...
6. (a) Let V be a vector space over the scalars F, and let B = (01.62, ..., On) CV be a basis of V. For v € V, state the definition of the coordinate vector [v]s of v with respect to the basis B. [2 marks] (b) Let V = R$[x] = {ao + a11 + a222 + a3r | 20, 41, 42, 43 € R} the vector space of real polynomials of degree at most three. Write down...
(d) Translate the following statement into predicate logic: “Every function f :R → R can be...
(d) Translate the following statement into predicate logic: “Every function f :R → R can be written as the sum of an even function and an odd function.” You can use the notation fi + f2 to represent the sum of functions fı and f2, and the notation f1 = f2 to represent the fact that fi and f2 are equal. 2n izo (e) Let n € N, and 20, 21, ..., Q2n E R. Let f: R + R...
Let Coo denote the set of smooth functions, ie, functions f : R → R whose nth derivative exists, ...
Let Coo denote the set of smooth functions, ie, functions f : R → R whose nth derivative exists, for all n. Recall that this is a vector space, where "vectors" of Coo are function:s like f(t) = sin(t) or f(t) = te, or polynomials like f(t)-t2-2, or constant functions like f(t) = 5, and more The set of smooth functions f (t) which satisfy the differential equation f"(t) +2f (t) -0 for all t, is the same as the...
Linear Algebra (5) Are the functions 1, 1, r in the vector space RR = {f:...
Linear Algebra
(5) Are the functions 1, 1, r in the vector space RR = {f: R R } linearly independent? Hint: Consider a linear combination of these functions and evaluate it at some specific points r = 0,1, ... to get several equations to solve for the coefficients. (6) Which of the following are bases of R3? Why or why not? 1-08 8 - 0 1 63.c-(07-08
(3) Let V denote a vector space over the field F and let v,..., Un E V. (a) Show that span(vn, 2,. , Un) (b) Show that span (ui , U2 , . . . , vn) span(v)+ +span(vn). span(v1)@span(v2)㊥·..㊥8pan(vn) if and only if (vi , . , . , %) is linearly independent.
Problem 6. Let Coo(R) denote the vector space of functions f : R → R such that f is infinitely differentiable. Define a function T: C (RCo0 (R) by Tf-f -f" a) Prove that T is a linear map b) Find a two-dimensional subspace of null(T).
2. (a) Let B = {f1, f2, f3} be a subset of P2 where fi(x) = x² – 3, f2(x) = x2 – 2x and f3(x) = x. Show that B is a basis of P2. (b) Determine whether or not the following sets are subspaces of F. (i) X = {f € F | f(x) = a(x + cos x), a € R}. (ii) Y = {f EF | f(x) = ax + sin x, a € R}. (c)...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
Problem 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...
6. (a) Let V be a vector space over the scalars F, and let B = (01.62, ..., On) CV be a basis of V. For v € V, state the definition of the coordinate vector [v]s of v with respect to the basis B. [2 marks] (b) Let V = R$[x] = {ao + a11 + a222 + a3r | 20, 41, 42, 43 € R} the vector space of real polynomials of degree at most three. Write down...
(d) Translate the following statement into predicate logic: “Every function f :R → R can be written as the sum of an even function and an odd function.” You can use the notation fi + f2 to represent the sum of functions fı and f2, and the notation f1 = f2 to represent the fact that fi and f2 are equal. 2n izo (e) Let n € N, and 20, 21, ..., Q2n E R. Let f: R + R...
Let Coo denote the set of smooth functions, ie, functions f : R → R whose nth derivative exists, for all n. Recall that this is a vector space, where "vectors" of Coo are function:s like f(t) = sin(t) or f(t) = te, or polynomials like f(t)-t2-2, or constant functions like f(t) = 5, and more The set of smooth functions f (t) which satisfy the differential equation f"(t) +2f (t) -0 for all t, is the same as the...
Linear Algebra
(5) Are the functions 1, 1, r in the vector space RR = {f: R R } linearly independent? Hint: Consider a linear combination of these functions and evaluate it at some specific points r = 0,1, ... to get several equations to solve for the coefficients. (6) Which of the following are bases of R3? Why or why not? 1-08 8 - 0 1 63.c-(07-08
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