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Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ]...
Prob 4. Consider a complex vector space Vspan (1, cos r, sinr, cos 2x, sin 2x) with an inner prod...
Prob 4. Consider a complex vector space Vspan (1, cos r, sinr, cos 2x, sin 2x) with an inner product fog@dt. Let U be the subspace of odd functions in V. What is U1? Find an orthonormal basis for both U and U
Prob 4. Consider a complex vector space Vspan (1, cos r, sinr, cos 2x, sin 2x) with an inner product fog@dt. Let U be the subspace of odd functions in V. What is U1? Find an orthonormal...
Let V = Cº(R) be the vector space of infinitely differentiable real valued functions on the...
Let V = Cº(R) be the vector space of infinitely differentiable real valued functions on the real line. Let D: V → V be the differentiation operator, i.e. D(f(x)) = f'(x). Let Eq:V → V be the operator defined by Ea(f(x)) = eax f(x), where a is a real number. a) Show that E, is invertible with inverse E-a: b) Show that (D – a)E, = E,D and deduce that for n a positive integer, (D – a)" = E,D"...
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...
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 f: R -R and g : R → Rbe some functions, and let x be a vector in R . Suppose that all the com...
Let f: R -R and g : R → Rbe some functions, and let x be a vector in R . Suppose that all the components off and g are directionally differentiable at x, and that g is such that, for all w RM, y +az) - g(y) y, w Then the composite function F(x)-g(f(x)) is directionally differentiable at x and the following chain rule holds: F, (x,d)=g'(f(x);f,(x,d)), YdER".
Let f: R -R and g : R → Rbe some...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Fi...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Find an orthonormal basis QWL of WL and find the dimension of WL 3. GIven a vector u- . find the Qw coordinate of Projw(v) . find the Qwa coordinate of Projwi (v) » find the coordinate...
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).
4. Let v={[a -.:a,nccc} Note that V is a vector space over R. View V as...
4. Let v={[a -.:a,nccc} Note that V is a vector space over R. View V as a R-vector space. (a) Find a basis for V over R. (b) Let W be the set of all matrices M in V such that M21 = -M12, where denotes complex conjugate. Show that W is a subspace of V over R and find a basis for Wover
#3 bullet 3 & #4 is denoted by llell and is calculated Note: The norm of...
#3
bullet 3 & #4
is denoted by llell and is calculated Note: The norm of a vector Consider a subspace W of R', W- span(v) Where 9-0-0 1. Find an basis Qw of W and find the dimension of W 2. Find an orthonormal basis Qwa of W and find the dimension of W 3, Given a vector u = find the w coordinate of Projw( find the Qw coordinate of Projw() find the coordinate of v in the...
(1 point) Are the following statements true or false? ? 1. The best approximation to y...
(1 point) Are the following statements true or false? ? 1. The best approximation to y by elements of a subspace W is given by the vector y - projw(y). ? 2. If W is a subspace of R" and if V is in both W and Wt, then v must be the zero vector. ? 3. If y = Z1 + Z2 , where z is in a subspace W and Z2 is in W+, then Z, must be...
Prob 4. Consider a complex vector space Vspan (1, cos r, sinr, cos 2x, sin 2x) with an inner product fog@dt. Let U be the subspace of odd functions in V. What is U1? Find an orthonormal basis for both U and U
Prob 4. Consider a complex vector space Vspan (1, cos r, sinr, cos 2x, sin 2x) with an inner product fog@dt. Let U be the subspace of odd functions in V. What is U1? Find an orthonormal...
Let V = Cº(R) be the vector space of infinitely differentiable real valued functions on the real line. Let D: V → V be the differentiation operator, i.e. D(f(x)) = f'(x). Let Eq:V → V be the operator defined by Ea(f(x)) = eax f(x), where a is a real number. a) Show that E, is invertible with inverse E-a: b) Show that (D – a)E, = E,D and deduce that for n a positive integer, (D – a)" = E,D"...
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...
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 f: R -R and g : R → Rbe some functions, and let x be a vector in R . Suppose that all the components off and g are directionally differentiable at x, and that g is such that, for all w RM, y +az) - g(y) y, w Then the composite function F(x)-g(f(x)) is directionally differentiable at x and the following chain rule holds: F, (x,d)=g'(f(x);f,(x,d)), YdER".
Let f: R -R and g : R → Rbe some...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Find an orthonormal basis QWL of WL and find the dimension of WL 3. GIven a vector u- . find the Qw coordinate of Projw(v) . find the Qwa coordinate of Projwi (v) » find the coordinate...
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).
4. Let v={[a -.:a,nccc} Note that V is a vector space over R. View V as a R-vector space. (a) Find a basis for V over R. (b) Let W be the set of all matrices M in V such that M21 = -M12, where denotes complex conjugate. Show that W is a subspace of V over R and find a basis for Wover
#3
bullet 3 & #4
is denoted by llell and is calculated Note: The norm of a vector Consider a subspace W of R', W- span(v) Where 9-0-0 1. Find an basis Qw of W and find the dimension of W 2. Find an orthonormal basis Qwa of W and find the dimension of W 3, Given a vector u = find the w coordinate of Projw( find the Qw coordinate of Projw() find the coordinate of v in the...
(1 point) Are the following statements true or false? ? 1. The best approximation to y by elements of a subspace W is given by the vector y - projw(y). ? 2. If W is a subspace of R" and if V is in both W and Wt, then v must be the zero vector. ? 3. If y = Z1 + Z2 , where z is in a subspace W and Z2 is in W+, then Z, must be...
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