We look at the inner product space of continuous functions above [0,1]. a) Calculate the angle...

We look at the inner product space of continuous functions above [0,1].
a) Calculate the angle between x and cosx ; x and sinx. Which angle is the smallest?
b) Calculate the distance between x and cosx ; x and sinx. What distance is the shortest? Hint: Remember that the distance between two vectors f and g is the length of f − g.
c) Sketch x, cosx and sinx in the interval [0,1]. Give a geometric explanation of the numerical values found in parts a) –b)

Expert Solution

Colby Messinger answered 2 months ago

Let W be an inner product space and v1, . . . , vn a basis...

Let W be an inner product space and v1, . . . , vn a basis of V . Show that <S, T> = <Sv1, T v1> + . . . + <Svn, T vn> for S, T ∈ L(V, W) is an inner product on L(V, W).

verify the assertion. (Subspace example) 1) The set of continuous real-valued functions on the interval [0,1]...

verify the assertion. (Subspace example) 1) The set of continuous real-valued functions on the interval [0,1] is a subspace of R^[0,1] This is from Linear Algebra Done Right- Sheldon Axler 3rd edition. I don't understand why the solution uses a integral.

Let C(R) be the vector space of continuous functions from R to R with the usual...

Let C(R) be the vector space of continuous functions from R to R with the usual addition and scalar multiplication. Determine if W is a subspace of C(R). Show algebraically and explain your answers thoroughly. a. W = C^n(R) = { f ∈ C(R) | f has a continuous nth derivative} b. W = {f ∈ C^2(R) | f''(x) + f(x) = 0} c. W = {f ∈ C(R) | f(-x) = f(x)}.

Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped...

Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped with the sup metric. Let Xi be the set of injective and Xs be the set of surjective elements of A and let Xis = Xi ∩ Xs. Prove or disprove: i) Xi is closed, ii) Xs is closed, iii) Xis is closed, iv) X is connected, v) X is compact.

Suppose V is a ﬁnite dimensional inner product space. Prove that every orthogonal operator on V...

Suppose V is a ﬁnite dimensional inner product space. Prove that every orthogonal operator on V , i.e., <T(u),T(v)> = <u,v>, ∀u,v ∈ V , is an isomorphism.

V is a subspace of inner-product space R3, generated by vector u =[1 1 2]T and...

V is a subspace of inner-product space R3, generated by vector u =[1 1 2]T and v =[ 2 2 3]T. T is transpose (1) Find its orthogonal complement space V┴ ; (2) Find the dimension of space W = V+ V┴; (3) Find the angle q between u and v; also the angle b between u and normalized x with respect to its 2-norm. (4) Considering v’ = av, a is a scaler, show the angle q’ between u...

(a). Check <u,v> =2u1v1+3u2v2+u3v3 is inner product space or not. If yes assume u= (8,0,-8) &...

(a). Check <u,v> =2u1v1+3u2v2+u3v3 is inner product space or not. If yes assume u= (8,0,-8) & v= (8,3,16) Find ||u|| ||v|| |<u, v>|2 Unit vector in direction of u and v Distance (u, v) Angle between u and v Orthogonal vectors of u and v. (b). Show that <u,v> =u1v1-2u2v2+u3v3

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be...

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be a subspace in R3 span by u = (9,2,0) and w=(9/2,0,2). a) Does V belong to W? show explanation b) find orthonormal basis in W. Show work c) find projection of v onto W( he best approximation of v with elements of w) d) find the distance between projection and vector v

Calculate the following inner product using distributed arithmetic principles: (-3, 5, 4)(0.5, -1.75, 0.25).

show that for any two vectors u and v in an inner product space ||u+v||^2+||u-v||^2=2(||u||^2+||v||^2) give...

show that for any two vectors u and v in an inner product space ||u+v||^2+||u-v||^2=2(||u||^2+||v||^2) give a geometric interpretation of this result fot he vector space R^2

Question