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

Expert Solution

Colby Messinger answered 1 year ago

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...

Let V be a vector space and let U and W be subspaces of V ....

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

(10pt) Let V and W be a vector space over R. Show that V × W...

(10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W and λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv. (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)

Let (V, ||·||) be a normed space, and W a dNormV,||·|| -closed vector subspace of V....

Let (V, ||·||) be a normed space, and W a dNormV,||·|| -closed vector subspace of V. (a) Prove that a function |||·||| : V /W → R≥0 can be consistently defined by ∀v ∈ V : |||v + W||| df= inf({||v + w|| : R≥0 | w ∈ W}). (b) Prove that |||·||| is a norm on V /W. (c) Prove that if (V, ||·||) is a Banach space, then so is (V /W, |||·|||)

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).

Let V be a vector space, and suppose that U and W are both subspaces of...

Let V be a vector space, and suppose that U and W are both subspaces of V. Show that U ∩W := {v | v ∈ U and v ∈ W} is a subspace of V.

6. Let V be the vector space above. Consider the maps T : V → V...

6. Let V be the vector space above. Consider the maps T : V → V And S : V → V defined by T(a1,a2,a3,...) = (a2,a3,a4,...) and S(a1,a2,a3,...) = (0,a1,a2,...). (a) [optional] Show that T and S are linear. (b) Show that T is surjective but not injective. (c) Show that S is injective but not surjective. (d) Show that V = im(T) + ker(T) but im(T) ∩ ker(T) ̸= {0}. (e) Show that im(S) ∩ ker(S) = {0}...

Questionnnnnnn a. Let V and W be vector spaces and T : V → W a...

Questionnnnnnn a. Let V and W be vector spaces and T : V → W a linear transformation. If {T(v1), . . . T(vn)} is linearly independent in W, show that {v1, . . . vn} is linearly independent in V . b. Define similar matrices c Let A1, A2 and A3 be n × n matrices. Show that if A1 is similar to A2 and A2 is similar to A3, then A1 is similar to A3. d. Show that...

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v....

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v. Suppose that ||u|| = 5 and ||v|| = 4. Find the cosine of the angle between w and v.

Let V be the vector space of all functions f : R → R. Consider the...

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B. b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One...

Question