Exercise 12. Let F, G: U C R2 → R' be a pair of mappings such that (F,G) = 0. Prove that (D,F, G) = -(F, D,G).

Q: (4) Suppose V and W are finite dimensional vector spaces such that dim(V) > dim(W). Show that any…

A:

Q: Let T be function from R to R defined as T (z) = (r, x), %3D Then T is not linear operator O True…

A: FALSE  the detailed explanation is as follows below:

Q: Let R = - b ,be Z}and let ø : R→Z be a mapping defined by : b = a-b. a a

A: We will find out the required value.

Q: Show that the function does not define an inner product on R3, where u = (u1, u2) and v = (v1, v2).…

A:

Q: 10. Let X be an inner product space and T: X→→→→→X an isometric linear operator. If dim X<∞, show…

A:

Q: Suppose {u1, Span{u;}nSpan{w;}={0}. Ur, W1, Ws} is a linearly independent subset of V. Show that •..…

A:

Q: Compute the Jacobian of the map G(u,v) = (u² + v²,u² – v²).

A:

Q: Let E The orthogonal projection of y on H = span{ u 1, ú 2} is given by:

A: Given

Q: Let T:V U be a homomprphism then ker T={0}if T is one-one.

A:

Q: If h:(R,+) → (R*,.) is a mapping defined by h(x)=10*, VXER, then (R,+) = (R*,.).

A:

Q: Let L : V W be a linear map. Let w e W. Let vo E V such that L(vo) solution of the equation L(X)= w…

A:

Q: Let T be a one-to-one linear transformation from a vector space V into Rn. Show that for u, v in V,…

A: Given that T be a one to one linear transformation from a vector space V into ℝn. Now the given…

Q: Let S={x | x=a+b +c 1 where a,b,c,deR. Show explicitly that S is a vector space.

A: We will construct a transformation matrix whose span will be the set S, and the  we will show that…

Q: [Exercise T1.13.] Prove that for any map f : R → R, a source has sensitive dependence on initial…

A: To Prove that for map f:ℝ→ℝ, a source has sensitive dependence on initial conditions:Firstly,Let x0…

Q: 2: Show that a map T is linear if and only if T(Au + v) = XT(u)+ T(v) for all A ER and vectors u, v.

A: A linear transformation satisfies  T:U-V  Then T(u+v) = T(u) +T(v)          T(au) = aT(u)  For all a…

Q: The mapping T : R² → R defined as T(u) = ||u||

A:

Q: Let U V be vector spaces and let T : U → V is a linear transformation prove that If {u1, u2, ·.. ,…

A:

Q: Let F,G be linear maps of vector space V onto itself. Show that: (FoG) = G -1 F-1,

A: Given that F,G be linear maps of vector space V onto itself.

Q: Evaluate f(x²y+3xyz) dxdydz by applying the transformation u = x, v=xy and w=3z, where G is G region…

A:

Q: Let V be an inner product space over R. Let u1, U2, Uz, U4 E V such that (u;, u;) < 0, i, j = 1, 2,…

A:

Q: Let R = =3a ala,be Z}and let o: R→Z be a mapping defined by: b = a-b. a

A:

Q: Let 7,V be a projection af the Vect or space. Accordling to this: Show that every Xe Im (T) for…

A: The given problem is related with projection. Given that T is a projection of the vector space V .…

Q: 5. Show that a translation t by a vector (a, b) defined by t(x, y) = (x + a, y + b) is an isometry.

A:

Q: The image of the line Im(z) = -2 under the mapping f(z)= i(z) is

A: Solution:

Q: let f (x, y) = x² + xy + y². Using the chain rule, calculate %3D af -(u(x, y), v(x, y)) being u = v…

A: Given that                                   f(x, y)=x2+xy+y2 And also we have…

Q: 7. Determine whether (u,v) =3u, v, – 2u, v, is a valid inner-product function or not for R²

A:

Q: Let T be the linear functional defined on C [a b] (with sup norm) by T (x) = ex (t) dt for x eC[a,…

A: Since T is a linear functional defined on        C [a,b]

Q: 3. Let C[0, 1] be equipped with the inner product (f, g) = f f(x) g(x) dx, where f = f(x) and g =…

A:

Q: Compute the norms for (a)||f(1)||, (b) ||F(x)|l2 (c)||f(x)||, where f(r) = -VI-r, defined on [0, 1).

A: Given- fx =-1-x defined on [0, 1] (a) we have to calculate fx  fx=∫01fxdx        =∫01-1-xdx…

Q: Let V and W be finite-dimensional inner product spaces. Let T: →W and U: W →V be linear…

A:

Q: Let T : R3 → R² be a linear mapping. 2 .T Given that T and T find T

A:

Q: Show that dim(W1) + dim(W2) – dim(W1n W2) = dim(W1 + W2), where W1 + W½ just denotes the span of W1…

A:

Q: The mapping w = z² + 1 is not conf ormal at O z = 0 O z = i O z = -i

A: We know that analytical function is not conformal at critical points. Let's find the critical points…

Q: ) For the vector space C[0,1], define the inner product (f,g) = , f(x)g(x)dx. Let f = x + 1 and g =…

A:

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

A:

Q: a Let R = . b DeZ and let o: R→Z be a mapping defined by: b = a-b. b a

A:

Q: Find the orthogonal projection of f onto g. Use the inner product in C[a, b] (f, g) = f(x)g(x) dx.…

A:

Q: Consider the spaces lP = P(N), 1 <p<00, i=0 For which pairs of (p, q), is the identity map I : x → x…

A:

Q: 2: Let a function d: R2 x R2 R defined by d(x, y) = ((x1 – y1)² + (x2 – y2)²)* %3D = XA Vx =…

A:

Q: Let V be an inner product space and T: V →V be a linear map. Suppose ||T (r) || = ||r||. Prove that…

A:

Q: Let T be a linear operator on an inner product space V. Let U1= T + T* and U2 = TT*. Prove that U1 =…

A: If T is an operator in an inner product space V over the field F together with an inner product,…

Q: Let H be the plane 2 X- 2 y- z=0 in R³, that is, H={(x, y, z)eR³ | 2x- zy- z=0} and let F be the…

A: Given that H=x,y,z∈R3|2x-2y-z=0 And, F(x,y,z)=2x-2y-2z2y+zy-z The aim is to find F(H).

Q: Find the orthogonal projection of f onto g. Use the inner product in C[a, b] (, 9) = [". f(x)g(x)…

A:

Q: Let G(u, v) = (5u + v, 4u + 4v) be a map from the uv-plane to the xy-plane. Find a point in the…

A:

Q: Let T be a linear transformation from V to W. Prove that the imageof V under T is a subspace of W.

A: Here given T be a linear transformation from vector space V to vector space W over the same field F…

Q: Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . Find the vector projection p of x onto y

A: the vector projection in the direction of another vector is the orthogonal projection of a vector…

Q: Let T: V→W be a linear transformation with vector spaces V and W both of dimension n. Then T is…

A: True

Q: §3.3, Exercise 16. Let P: R" → R™ and Q : R" → R™ be linear mappings. Prove that S: R" → R" defined…

A:

Q: Suppose that U is a linear transformation from R" into R" that is isometric, meaning that ||Ux|| =…

A: Let U: Rn → Rm be the isometric transformation and Ux=x for all x∈Rn (a) We have to prove that…