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9. a) Let W be the set of continuous functions in C(-00, such that jr (*)dx...
(1) Let C[a, b] denote the set of functions that are continuous on (a,b). Define a...
(1) Let C[a, b] denote the set of functions that are continuous on (a,b). Define a function on pairs of function fig E C[a, b] by dı(,9) = /\f(x) – 9(2)| dx. Prove that (C[a,b], du) is a metric space.
Let Let B = {e3x, xe, x2e5x} be a basis for subspace W of the set...
Let Let B = {e3x, xe, x2e5x} be a basis for subspace W of the set of continuous functions, and let D, be the differential operator on W. a) Find the motrix for D, relative to hosts B b) Use the matrix to evaluate D(45+ – 6xe** + 5x225)
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
12. Let M be the set of continuous functions on R which vanish outside a finite...
12. Let M be the set of continuous functions on R which vanish outside a finite interval (the interval may depend on the function). (a) Show that M is a metric space in the sup norm. (b) Show that M is not complete. Chapter 5. Sequences of Function 210 (C) Show that C.(R), the continuous functions which go to zero et is complete in the sup norm (problem 10 of Section 5.3). (d) Prove that Mis dense in Co(R). T...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be the set of...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
6. Let W be the set of all vectors of the form W {(a,b,c): a –...
6. Let W be the set of all vectors of the form W {(a,b,c): a – 2b + 4z = 0} Is W a subspace of the vector space V = R3?
Let W denote the set of smooth functions f(x) in CⓇ such that f"(x) = -f(x)....
Let W denote the set of smooth functions f(x) in CⓇ such that f"(x) = -f(x). That is, W = {f(x) in "S"(t) = -f(x)} . W is a subspace of C . For all a and b, a sin(x) + bcos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b) Let's say we knew that dim(W)=2. Show that (sin(x),cos(x)} is...
(5) Let W denote the set of smooth functions f(2) in CⓇ such that f'(x) =...
(5) Let W denote the set of smooth functions f(2) in CⓇ such that f'(x) = -f(L). That is, W= {f() in C | F"(x) = -f(x)} In the previous worksheet, we showed that: • W is a subspace of Cº. . For all a and b, a sin(2) + b cos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b)...
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For a...
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For any number ce [0, 1] (a) Show that the set R is a ring and that the set Ic is an ideal of R. (b) Is I UI2 and ideal? Is I, nI an ideal?
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval...
(1) Let C[a, b] denote the set of functions that are continuous on (a,b). Define a function on pairs of function fig E C[a, b] by dı(,9) = /\f(x) – 9(2)| dx. Prove that (C[a,b], du) is a metric space.
Let Let B = {e3x, xe, x2e5x} be a basis for subspace W of the set of continuous functions, and let D, be the differential operator on W. a) Find the motrix for D, relative to hosts B b) Use the matrix to evaluate D(45+ – 6xe** + 5x225)
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
12. Let M be the set of continuous functions on R which vanish outside a finite interval (the interval may depend on the function). (a) Show that M is a metric space in the sup norm. (b) Show that M is not complete. Chapter 5. Sequences of Function 210 (C) Show that C.(R), the continuous functions which go to zero et is complete in the sup norm (problem 10 of Section 5.3). (d) Prove that Mis dense in Co(R). T...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
6. Let W be the set of all vectors of the form W {(a,b,c): a – 2b + 4z = 0} Is W a subspace of the vector space V = R3?
Let W denote the set of smooth functions f(x) in CⓇ such that f"(x) = -f(x). That is, W = {f(x) in "S"(t) = -f(x)} . W is a subspace of C . For all a and b, a sin(x) + bcos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b) Let's say we knew that dim(W)=2. Show that (sin(x),cos(x)} is...
(5) Let W denote the set of smooth functions f(2) in CⓇ such that f'(x) = -f(L). That is, W= {f() in C | F"(x) = -f(x)} In the previous worksheet, we showed that: • W is a subspace of Cº. . For all a and b, a sin(2) + b cos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b)...
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval [0, 1] to the real numbers. For any number ce [0, 1] (a) Show that the set R is a ring and that the set Ic is an ideal of R. (b) Is I UI2 and ideal? Is I, nI an ideal?
C(10.1]) be the set of continuous functions f : lo. 11 → R 5) Let R from the interval...
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