1. Show that the following maping f are linear: f : R² → R² defined by f(x, y) = (x+ y,x).

Q: 4) For which one of these linear maps T:R → R? is det T > 0? a) T(1, y) = (y, -r) b) T(1, y) = (x,…

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Q: Is the map T: R² → R² given by T(x, y) = (x,y + 1) linear?

A: T: V-> W  mapping from V to W is a linear map if and only if  ->(u+v)=T(u)+T(v) ->T(au) =…

Q: Suppose the motion of a continuous body is given by x(X, t) = (X1 +tX2)en +(X2-tX1)ez + X3e3…

A: The motion described is by linear combinations. We can write the linear combination in the form of a…

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: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = √(x2 + y2 +…

A: Given: Function f(x,y,z) as follows: and a point (1,2,2) To find: Linearization L(x,y,z) of the…

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: 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: (a) Show conclusively that the map f : R² → R defined by f(x, y) = x cos(y) is not linear.

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Q: Which of the following maps from R³ to R³ are not linear? f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

A: The function f is said to be linear if: f(X+Y)=f(X)+f(Y)f(kX)=kf(X) where X,Y∈ℝ3.

Q: Given the map f:R-R+ defined by f(x.y.z) = (0,0,x.0) then nullity of f is: O A.4 O B. 3 OCO O D-2 OE…

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Q: Which of the following maps from R³ to R³ are not linear? U □ f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

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Q: Define f : R 2 → R by f(x, y) = x 2 + y 2 . Compute the linearization of f at (−1, 1).

A: We have to find the linearlization at (-1, 1)

Q: The domain of the function f (x, y, z) = In(6 - 3xy) %3D is {(x, y) E R? : xy 2} {(x, y, z) E R :…

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Q: Given the map f:R3-R4 defined by f(x,y,z) = (0,0,0,0) then nullity of f is: OA. 3 O B. 4- OC.O O D.2…

A: For the solution of the problem follow the next step.

Q: Please solve this question.

A: We are given that L(p(x)) = p(0). Let p(x), q(x) in P4(F) and α, β in the field F. Now,…

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = x2 + y2 + z2…

A: given that a function f(x,y,z)=x2+y2+z2 we need to find the linearizations of f(x,y,z) at (1,1,1).…

Q: of at (x, y, z) = (2, 3, 4). ay Let f(x, y, z) = (x + y)z. Evaluate

A: Given :fx,y,z = x+yz

Q: Which of the following maps is not linear? А. T:R²→R² defined by T(x,y) = (x²2,x+y)· В. f:R3→R2…

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Q: 2. Let L: V W be a linear map. Let w E W. Let vo E V such that L(vo) = w. Show that any solution of…

A: Solution is in step 2

Q: Determine the Wronskian of the given set of functions and determine whether they are linearly…

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

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Q: Let f, g, ϵ F (R , R) be the functions defined by f (t) = ert and g(t) =est, where    r ≠ s. Prove…

A: We can prove this question by the contradiction method, So, Assume that the opposite. Let {f, g}…

Q: Are the following linear maps invertible? a) T: R3 → R³ defined by T(x, y, z) = (y, z, x)?

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Q: Let f(r, y) = |r|+ lyl- (a) Draw three r-cuts of f(r, y), three y-cuts of f(x,y), and one z-cut.…

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Q: then Given the map f: R³→R4 defined by f(x,y,z) = (0,0,x,0) rank of f is: А. В. 3. Е. 2. 4-

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Q: Let f(x, y) = x² + In(2x – y). Find the gradient Vf(x, y) at the point (1, 1). O O O O

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Q: Given the map f:R3 R4 defined by f(x,y,z) = (0,0,0,0) then rank of f is:

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

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Q: (b) Let T : V → R be a linear map. If Im(T) = {0}, show that T(T) = (T, 0) where õ e V.

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Q: Let f(x, y) = x2 − 2y and g(x, y) = xey . Show that at any point (x0, y0), the level curve of f…

A: To show that at any point the level curve of f and g intersects perpendicular, it is enough to show…

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: Let A = {u, v, w}, B = {x, y, z}. Determine whether the relation R = {(u, x), (v, z), (w, y)} from A…

A: In R from A to B is an injection function as well as a surjection function.

Q: Let T be function from R to R2 de fined as T (x) = (0, x), Then show that if T is linear operator or…

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Q: Let f: R → R be the function which satisfies for all (z. y, z) € R° that z cos(ry) and let C C R' be…

A: Here we have to find out line integral of function f(X,Y,Z).

Q: Suppose we have a surface f (x,y)=x²y+cos (4x) · Which of the following is the gradient of f(x,y) at…

A: First I have found the partial derivatives of f(x,y)  with respect to 'x' and 'y' at point P(π/2,…

Q: suppose that x,y,z∈R3 are such that x.(y×z)≠0. prove that x,y,z are linearly independent.

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Q: Which of the following maps is not linear? O A. L-R2-R3 defined by L(x,y) = (x+y,y.x)- OB. K:R R3…

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Q: 1 Are the following sets of functions, each defined on x E R, linearly independent? Use Wronskian.…

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Q: Let f(z) = u(x, y)+iv(x, y) be entire such that u(x, y) = 4v(x, y). Show that f'(z)=0

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Q: PROVE: If T is a linear map from V to W, then T(0) = 0.

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

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Q: For the map F: R³ → R² defined by F(x, y, z) = (x²y + e², yz − x)

A: Given,            F(x, y, z)=x2y+ez, yz-x

Q: Let F(x, y) = <3x?y3 + y", 3x°y² + y^ + 4xy³,. %3D Evaluate F. dr where C is the line segment from…

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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 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…

Q: Let T : R³ → R³ be a surjective linear map. What can you say about the dimension of ker(T) ?

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Q: Let T be function from R to R² de fined as T (x) = (x, 0), Then show that if T is linear operator or…

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Q: Use the Wronskian to show that the given functions are linearly independent on the given interval I.…

A: For linearly independent, we show Wronskian is non -zero.