Determine whether each of the following maps from R? to R³ is linear. Prove your answers. (a) F((x, y))= (æ², y², x + y) (b) G( (x, y)) = (x+y, 2æ, 3y) (c) H((x,y))= (x +1, y + 2,0)

Q: Let f(x, y) = V2x+4y+32-X. The domain of C for which the level curve f(x, y)=c is an ellipse i O…

A:

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: Find the mapping that reflects the plane in the line x+y-1=C O (x,y)>(y+1,x+1) (x,y)→(-x+1,-y+1) O…

A:

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

A:

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

A:

Q: Which of the following maps T, L, I, f: R³ → R³ is not linear? a) 1(x, y, z) = (x, y, z). b) f(x, y,…

A:

Q: Verify that the following pairs of curves meet orthogonally x2 + y2 = 4, x2 = 3y2

A: The slope of the tangent of x2 + y2 = 4  is  The slope of the tangent of x2 = 3y2  is

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…

A:

Q: Let H be the plane Зx-3у+2z-0 (*) in R³, that is, H={(x, y, z)OR³ | 3x-3y+2z=0} and let F be the…

A: Given, H ={(x, y, z)∈ℝ3| 3x-3y+2z=0}Since, 3x-3y+2z=0⇒x =3y-2z3⇒H =3y-2z3, y, z∈ℝ3| 3x-3y+2z=0…

Q: The functions y1, Y2, . .. , Yn are linearly independent, if and only if W (y1, 32, ..., Yn)(to) #0…

A: If two functions are linearly dependent then the Wronskian is zero for every value of t.

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) =…

A:

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 :…

A:

Q: The set of functions {Vx, x, x²} is linearly dependent on (0, oo). True False

A:

Q: 2. - For each map below, either prove that it is linear or demonstrate that it is not linear. (a) T…

A: Consider a map T:U→V is linear if    (i) T(u+v) =T(u)+T(v)   (ii) T(αu )   = αT(u)   , for all u,v…

Q: Question 1. Are the following linear maps invertible? a) T: R³ → R³ defined by T(r, y, z) = (y, z,…

A:

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

A:

Q: Let f(x, y) = 2x² + 5xy + 2y² + 5x – 5y + 2. Find the linearization L(x, y) of f(r, y) at the point…

A:

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: Find the linearization of f(x, y,z) = x + y centered at (8,4, 5).

A: Given:fx,y,z=x+y at 8,4,5Formula:Lx,y,z=fx0,y0,z0+fxx0,y0,z0x-x0+fyx0,y0,z0y-y0+fzx0,y0,z0z-z0

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

A:

Q: Let f (x, y) = x² – y² . Find all all points (æ, y) such that Vf (x, y) = ±(1, 1).

A: givenf(x, y) =x2-y2∇f(x, y) =±(1, 1)

Q: Consider the piecewise smooth curve C = C1 u C2 u C3 u C4, oriented from the point A(1, 1, 2) to…

A: Given that, F=4x3-2z, 3y2,-2x Find the value of ∫cF. dr we have to find the function f such that…

Q: Let T: R2 R be a linear map such that T(x, y) = (2x, 3y, 2y- r), then one of the following in range…

A:

Q: Draw a contour map for f(x,y) = x^2 + 4y^2 where k = 3, k=8 and k=14

A: A scalar function f(x,y) is a rule that is used to associate a number, or quantity with a particular…

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

A:

Q: find the linearization at x = a. y = (1 + x2)^−1/2, a = 0

A: We find the derivative of the given function using power rule and chain rule

Q: 2. Show that the equations f(r, y,P. q) = 0 and g(r, Y.P. q) = 0 are compatible if %3D %3D a(f,g)…

A:

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

A:

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: If L(x, y) is the local linear approximation of f(x, g(2, 1)= 4, g,(2, 1)=4 and g,(2, 1)=6, then L O…

A: Use formula of linear approximation

Q: Question 1. Are the following linear maps invertible? a) T : R³ → R³ defined by T(r, y, z) = (y, z,…

A: A mapping is sid to be linear transformation if, (1) T(cx+y)=cT(x)+T(y) for x, y any vector and c be…

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…

A:

Q: Let T : R² → R³ be the linear map given by [2r1 + 7x2] 2i + 22 Write does the matrix A associated…

A: The given linear map is T:ℝ2→ℝ3Tx1,x2=2x1+7x2, x1-x2, x1+x2 To find the matrix A associated with T…

Q: a +c b+c b) h(a + bx + cx?) = %3D 1+b+c, Is this mapping linear?

A: Given : h(a+bx+cx2)=a+cb+c1+b+c To Find : Determine the given mapping is linear.

Q: Consider the linear map T: M, ,- R° defined by 2a+b a b c d 2c+d Find either the nullity or the rank…

A:

Q: Let L:R R be a linear map defined by L(x,y.2) = (x+ 2y 2r+ 3y+ z,x+ y+2) Then what is the image of ?…

A:

Q: Question 18 Which of the following maps is not linear? T.L.I.f:R3-R3 O a. T(x.y.z) = (1,0,0): Ob. f…

A:

Q: Question 13 Which of the following maps is not linear? O A T:R2→R2 defined by T(x,y) = (x²,x+y)* O…

A: Option A is not linear map because it does not satisfy linear property. All other satisfy linear…

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

A: Solution is given below

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: J f ds if Evaluate Of(x, y, z)=x+y, and C is the combinations of lines connecting adjacently the…

A: As per company guidelines we are allowed to solve one question at a time so i am solving first one…

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: Which of the following maps from R³ to R³ are not linear? 0 f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

A: The solution is given below

Q: Find the linearization L(x,y,z) of the f(x,y,z)= 8 square(x^3+y^3+z^3) at the point (1,2,3)

A:

Q: Letf(x, y.z) be a function with the gradient vector . What are the extreme points of f with…

A: The given constraint can be written in the form of g(x,y,z)=0 as shown below:…

Q: State whether the following statement is true or false: If f:U V is a linear map, then Kerf + Ø O…

A: kernel of a function is the set of all elements which map to 0.