Find the rank of the mapping h : P4 → P4 defined by h(a + bx + cæ² + dæ³ + ex*) = 2c + 6dx + 24ex²

Q: Let T: R2 - R3 be the linear map which satisfies and 3 Give the formula for T. x + 3y x + y

A: We have to find

Q: For z € C, define Tz: CC by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Given that: For z∈C, Define Tz:C→C by Tzu=zu. The objective is to verify the given statement. Now…

Q: Given the map f:R→R4defined by f(x.y.z) = (0,0,0,0) then nullity of is: O A. 4 OB 2 OC1 OD.3

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Q: Let →a=⟨3,1⟩ and →b=⟨1,0⟩. Find the projection of →b onto →a. proj→a →b=

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Q: In each case find the norm of the linear map L directly from the definition: (a) L : R → R with L(x)…

A: (a)  Given that the linear transformation L:ℝ→ℝ   with  Lx=-2x. The norm of the linear…

Q: Suppose M and N are C manifolds, U an is C. Show that there exists a neighborhoo that F can be…

A: Let us start solving the problem step by step

Q: Show that the map T : R3 -→R? given by T (x1, X2, X3) = (x1 + x2, X3) is an open map.

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Q: Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x+ azx² + a3x³) = ao +…

A: For T to be an isomorphism we must check that it is one one and onto both. And we can check it by…

Q: Prove that the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.

A: A multiplication operation defined on ℤ/nℤ×ℤ/nℤ→ℤ/nℤ as                x·y=xy

Q: In which of the following cases is the mapping x: R² → R³a patch? (а) x(и, v) — (и, uv, v). (c) x(и,…

A: A mapping defined from x : R3 to R2 is defined a patch if x maps U diffeomorphic ally onto x (U) As…

Q: 16. Let P : R" → R" and Q : R" → R™ be linear mappings. (a) Prove that S: R" → R™ defined by S(x) =…

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Q: Determine if the ODE is a homogeneous DE. If it is, state its degree. 1. у' 2ху2 yx2 +y3 -4/(x³y3)

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

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Q: Let T : V → U be a homomorphism, then Ker T = { 0 } iff T is one-one

A: A linear transformation is said to be injective or one to one if provided that for all v1 and v2 in…

Q: Consider P2 with the inner product (p, q) = | | p(x)q(x) dx . Let U = span{1, 2x + 2} and g(x) = 3x2…

A: See the detailed solution below.

Q: Define a map t: R × R → R × R by t(a, b) = (a+ b, a - b). Prove that t is a one-to-one…

A: The solution is given as

Q: Given the map FR4R3 defined by f(x,y.z.w) = (0,0,0). Then rank of O A. 1- OB. 4 OC.O OD. 3 O E. 2

A: Given map is f:ℝ4→ℝ3  defined by f(x,y,z,w)=(0,0,0). we have to find the rank of f. The map f is…

Q: Let B = {(p,q) e R² : p² + q² = 1}. (a) Show that B is NOT disconnected (b) Given that there is no…

A: Given: B=p,q∈ℝ2 : p2+q2=1. There is no continuous mapping f:B→ℝ. To prove: B is not disconnected.…

Q: 2. Find the complement CC of the set C with respect to the space C if (a) C = {x :0 < x < 1}, C = {x…

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Q: Let T : P4 P4 be linear map such that T(p) = p", then rank(T) = %3D A) 4 В) 3 С) 2 D) 5

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Q: 3) Prove that the multiplication map · : Z/nx Z/n → Z /n given by [x] · [y] = [xy] is well-defined.

A: To prove that the multiplication map · : ℤn×ℤn→ℤn given by    x·y=xy  is well-defined.

Q: Determine the image of the line Im(z)= -2 under the mapping f(z)= i(z ) and describe it in words.

A: Im(z)=-2 represents a straight line y=-2let z=x-2i where x is real number. ⇒u+iv=f(z)=iz2=ix+2i2…

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

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Q: Let p: GL(2, R) → R' be a homomorphism map defined by p(A) = det(A), then %3D a) Ker o = GL(2, R) b)…

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Q: Let v e R" and letk e R. Prove that S = {x € R" :x•v = k} is an affine subset of R". %3D

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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: Given the map f:R4 3 defined by f(x,y,Z,w) = (0,0,0) Then rank of , is: f O A. 1 O B. 3 OC.O O D.2 O…

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Q: 9. Let F : R" → R". Prove that for each a the mapping h→ dF[a; h] is linear on R".

A: Given: F:ℝn→ℝm   To prove: For each a, the mapping h↦dFa;h is linear on ℝn

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

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Q: is any linear map, then the foll Im f¹ = (Ker (f))º Ker (f¹) = (Im f)⁰ T0

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Q: 4. Let L: R? H R? be the linear map defined by L(r, y) = (x + y, x - y). Show that L is invertible.…

A: Given problem is :

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: (H.W): Find (D) & (Rf) for this mapping F:Z-Z, f(x) = x² %3D

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Q: Every continuous map T2 –→ S2 is homotopic to the constant map

A: To prove: Every continuous map from the Torus T2 to the sphere S2 is homotopic to the constant map.

Q: Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1r + azx² + azx³) = ao +…

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Q: Let L be the line in R2 through the points [:][] and Find a linear functional fand a real number d…

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Q: Show that the functions x and x2 are orthogonal in P5 with inner product defined by (5), where xi =…

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Q: Suppose T E L(U,V) and S € L(V, W) are both invertible linear maps. Prove that ST E L(U, W) is…

A: First we prove that ST is a invertible linear map from U to W and then it will be shown that the…

Q: Find det (T) for T:R- R, where T(x,,×2,*3) = (x,- x2, X2 - X3, X3-x).

A: We will find the matrix [T] from the image of the standard basis vectors of R³ under the…

Q: Find the image of |z + pi + 2p| = 4 under the mapping w = p√² (e¹²) z. about

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Q: Find all maps ¢:(Z,+) → (Q*, ×) satisfying O (a + b) = ¢(a) × ¢(b) Va, b e Z

A: We derive the function from the given properties.

Q: Consider the map S: M2,2 → M22 defined by S(A) = AA.

A: Consider the map S(A): M2,2→M2,2ByS(A)=AATo check S(A) is linear transform.

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

A: Ring homomorphism

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…

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Q: Compute the Jacobian of the mapping. G(r, t) = (2re", 3 + 8e")

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Q: (3) Suppose that T: V F is a linear map. Prove that if u E V such that u 4 Ker(T) then V = Ker(T)…

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Q: Let L and M be two continuous linear maps such that the composite Mo L is defined. By quoting…

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Q: Find the image of | z – 3i | = 3 under the mapping W =-.

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Q: Let T : R? →→ R² and S : R² → R² be a linear maps such that r() - (1) (G) = (;) TO and S(( 2 1 Then…

A: Given: T: R2→R2 ,  S: R2→R2 be linear maps such that T10=11 &amp; S11=12 To determine: S∘T10

Q: Let L : R? → R? be the linear map defined by L(x, y) = (x + y, x – y). Show that L is invertible.…

A: Here given that L :ℝ2→ℝ2 be the linear map defined by L(x, y)=(x+y, x-y). We have to show that L is…