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7. Let V be the space generated by the basis B = {sin(t), cos(t), et}. i.e....
2. Consider the inner product space V = P2(R) with (5,9) = . - f(t)g(t) dt,...
2. Consider the inner product space V = P2(R) with (5,9) = . - f(t)g(t) dt, and let T:V + V be the linear operator defined by T(F) = xf'(x) + 2f (x). (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]2 is diagonal. If such a basis exists, find one.
2. Consider the inner product space V = P2(R) with (5.9) = £ 5(0)9(e) dt, and...
2. Consider the inner product space V = P2(R) with (5.9) = £ 5(0)9(e) dt, and let T:V V be the linear operator defined by T(f) = xf'(2) +2f(x). (i) Compute T*(1+2+x²). (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
4 Let T:R? R'be the operator defined as T(v) = (v. du for a fixed unit...
4 Let T:R? R'be the operator defined as T(v) = (v. du for a fixed unit vector (cos , sin). Find the simple matrix for T in rotated coordinates and then transform back to standard coordinates to find 17 We were unable to transcribe this image
5. Let V be the subset of Cº(R) consisting of all functions that can be expressed...
5. Let V be the subset of Cº(R) consisting of all functions that can be expressed in the form a sin 2x + be 4x + cos2r for a, b, c ER. (a) (4 points) Prove that V is a subspace of C(R). (b) (3 points) Let fix) = sin 2x + e4r f2(x) = sin 2.c + cos 2.0 f3(x) = 4x + cos2r. The set B = (f1, f2, f3) is an ordered basis for V. (You do...
3.2.1: Let V be a vector space with Basis B and let L be an operator...
3.2.1: Let V be a vector space with Basis B and let L be an operator on V. L2 means the operator applied twiceL2(v) = L(L(v)). Show that the Matrix of L2 is the square of the matrix of L, i.e LL? Demonstrate this problem for the space V span (1, t, t2, t3) and let L d/dt (so L2d/dt) SO
please help me with questions 1，2，3 1. Let V be a 2-dimensional vector space with basis...
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
Let V be the vector space consisting of all functions f: R + R satisfying f(x)...
Let V be the vector space consisting of all functions f: R + R satisfying f(x) = a exp(x) +b cos(x) + csin(x) for some real numbers a, b, and c. (The function exp refers to the exponential, exp(22) = e.) Let F be the basis (exp cos sin of V. Let T :V + V be the linear transformation T(f) = f + f' + 2f" (where f' is the derivative of f). You may use the linearity of...
Let B = {b1,b2, b3} be a basis for a vector space V. Let T be...
Let B = {b1,b2, b3} be a basis for a vector space V. Let T be a linear transformation from V to V whose matrix relative to B is [ 1 -1 0 1 [T]B = 2 -2 -1 . 10 -1 -3 1 Find T(-3b1 – b2 - b3) in terms of bı, b2, b3 .
1. Let B { 1, х, хг} et S {x2 +x, 2-1, x+1 } be two...
1. Let B { 1, х, хг} et S {x2 +x, 2-1, x+1 } be two basis of P2. Let T : P2 P2 be a linear transformation such that 3,S 2 2 -2 Find a basis of Ker(T), a basis of Im(T) and T^b 2. Let Let : P1 → P1 be a linear transformation such that 4 -3 where B-[1, x,} et S - {2c - 1,x - 1} be two basis of P1. Find A2 and T2....
Consider the inner product space V = P2(R) with (5,9) = { $(0)g(t) dt, and let...
Consider the inner product space V = P2(R) with (5,9) = { $(0)g(t) dt, and let T:VV be the linear operator defined by T(f) = x f'(x) +2f (x) +1. (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]k is diagonal. If such a basis exists, find one.
2. Consider the inner product space V = P2(R) with (5,9) = . - f(t)g(t) dt, and let T:V + V be the linear operator defined by T(F) = xf'(x) + 2f (x). (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]2 is diagonal. If such a basis exists, find one.
2. Consider the inner product space V = P2(R) with (5.9) = £ 5(0)9(e) dt, and let T:V V be the linear operator defined by T(f) = xf'(2) +2f(x). (i) Compute T*(1+2+x²). (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
4 Let T:R? R'be the operator defined as T(v) = (v. du for a fixed unit vector (cos , sin). Find the simple matrix for T in rotated coordinates and then transform back to standard coordinates to find 17 We were unable to transcribe this image
5. Let V be the subset of Cº(R) consisting of all functions that can be expressed in the form a sin 2x + be 4x + cos2r for a, b, c ER. (a) (4 points) Prove that V is a subspace of C(R). (b) (3 points) Let fix) = sin 2x + e4r f2(x) = sin 2.c + cos 2.0 f3(x) = 4x + cos2r. The set B = (f1, f2, f3) is an ordered basis for V. (You do...
3.2.1: Let V be a vector space with Basis B and let L be an operator on V. L2 means the operator applied twiceL2(v) = L(L(v)). Show that the Matrix of L2 is the square of the matrix of L, i.e LL? Demonstrate this problem for the space V span (1, t, t2, t3) and let L d/dt (so L2d/dt) SO
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
Let V be the vector space consisting of all functions f: R + R satisfying f(x) = a exp(x) +b cos(x) + csin(x) for some real numbers a, b, and c. (The function exp refers to the exponential, exp(22) = e.) Let F be the basis (exp cos sin of V. Let T :V + V be the linear transformation T(f) = f + f' + 2f" (where f' is the derivative of f). You may use the linearity of...
Let B = {b1,b2, b3} be a basis for a vector space V. Let T be a linear transformation from V to V whose matrix relative to B is [ 1 -1 0 1 [T]B = 2 -2 -1 . 10 -1 -3 1 Find T(-3b1 – b2 - b3) in terms of bı, b2, b3 .
1. Let B { 1, х, хг} et S {x2 +x, 2-1, x+1 } be two basis of P2. Let T : P2 P2 be a linear transformation such that 3,S 2 2 -2 Find a basis of Ker(T), a basis of Im(T) and T^b 2. Let Let : P1 → P1 be a linear transformation such that 4 -3 where B-[1, x,} et S - {2c - 1,x - 1} be two basis of P1. Find A2 and T2....
Consider the inner product space V = P2(R) with (5,9) = { $(0)g(t) dt, and let T:VV be the linear operator defined by T(f) = x f'(x) +2f (x) +1. (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]k is diagonal. If such a basis exists, find one.
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