Let T : R → R be the linear operator given by T(a, b, c) = (a, 2b, 3c) . Then det T = O5. O 2. 6. O 1. 4.

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 T be the linear operator on R defined by T(x,,x, x) = (2x1, x, - x, 5x, + 4x, + x,). Then 7 is…

A: First express the linear operator in matrix form and then use the matrix to calculate the inverse…

Q: Consider the following linear operator; T : R → R° given by T(x, y, z) = (y, x + z, z).

A: We have to find the real Eigen value of T

Q: Prove Theorem (A + B )T = AT + B T

A: Prove Theorem (A + B )T = AT + B T

Q: 3. Let T1 : R2 → R² and T2 : R → R² be the linear operators given by the formulas T:(1, y) = (r + y,…

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Q: Let V be a vector space, T a linear operator on V, and v, w E V. Suppose that T(v + w) = v – 2w and…

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Q: If |r(t)| = c (a constant), then r' (t) is orthogonal to r(t) for all t.

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Q: Let L:R R be a linear operator such that L X2 = X1 then Ker(L) spanned by: -> Select one: a. (e2,e3)…

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Q: Q: Let T: R² → R² be given by T(x, y) = (x, −y). Then show that T is a linear operator.

A: Solution

Q: Consider the functions f(x, y, z, t) z?, t?) and g(u, v) = (uv, u +v, u – v). Use the chain rule to…

A: I have solved in general and also using chain rule. Composition  of several variable functions:  Let…

Q: et X be a normed linear space and T: X→ X and S: X→ X be two perators, then S T :X→X is a bounded…

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Q: Let w = r = Uv, s = u - 3v. Use appropriate forms of the + s²' dw dw chain rule to find and in terms…

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Q: Let T₁ : M₂2 · R³ and T₂ : R³ → R4 be defined by T₁ ([a b]) = = = (a, 2b, 3c) T₂(x, y, z) = (x, y,…

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Q: Determine whether the following are linear operatorson Rn×n. L (A) = A + I

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Q: Let T: R² R2 be a linear operator defined by Find a formula for T I Y T[H]-[2] + [2] = [3]. 5

A: To find the formula for the linear transformation of [x y].

Q: Consider the D-operator P(D) = mD² + 2kD +¢, where m and c are positive. If -c/m < k< 0, then

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Q: Let L:R→ R be a linear operator such that L X2 )= X1 then Ker(L) spanned by: X3 X1 Select one: a.…

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Q: If S and T are normal linear operators satisfying ST* = T* S and TS* = S*T, show that their sum S+T…

A: I have used just definition of normal operator

Q: Let T be the function that maps R? into R? such that T(u) = proj,u where v = (1, 1). (a) Find T(x,…

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Q: Let LR R be a linear operator such that L X, )-| X1 then Ker(L) spanned by: Select one: a. (e2,e3)…

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Q: Let T1 : R" → R* and T2 : R* → R" be linear transformations. Prove that for all k e R, and ve R":…

A: Idea: using definition of linear transformation  T(kv)=kT(v)

Q: Let T: X→→→→→ Y be a linear operator and dim X = dim Y=n<∞. Show that R(T) = Y if and only if T¹…

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Q: Ques. 3: Check whether the operator T: R2 → R² defined by T(x,y) = (y,x) is linear. Moreover if T is…

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Q: Say the linear operator given T:R³ → R³, defined by T(x, y, z) = (x + y, x – y + 2z, 2x + y – 2). I…

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Q: Let T: R-R° be a linear operator given by T(x, y, z) = (x+y, x-y, 0) which of the following vector…

A: You have attached question 8 only but above thta you mention Q7.I am answering question 8.

Q: Let T be the function that maps R2 into R2 such that T(u) = projvu, where v = (1, 1).(a) Find T(x,…

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Q: Show that the operators T,··, T4 from R² into R² defined by ... (§i, 52)→ (Š1,0) (§1, É2)→ (0, E2)…

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Q: 2. Let T: R³ R³ be the operator given by 1 1 T(v) = 9). 1 Determine whether T is decomposable or…

A: Consider the given operator, Tv=-1100-1100-1v Here we need to determine that whether T is…

Q: a bonneled linear operator. Then prve. that. Kes's is clased. b) let 8:X>y be a

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Q: Describe all linear operators T on R2 such that T is diagonalizable and T3−2T2 + T = T0.

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Q: Consider the functions f(x, y, z, t) = (x² + 2. t²) and g(u. v) = (uy, u +v. u – v). Use the chain r

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Q: Question 5. Let T : V → V be a linear operator satisfying T² = T. Define U1 = {v € V : T(v) U2 = {v…

A: Given: U1=v∈V:Tv=v U2=v∈V:Tv=0 T2=T Objective: To show that V=U1⊕U2 That means to show that every…

Q: A E R"X", Q e R"Xn and Q > 0, C e R™×n. For a given X € R"Xn and X > 0, define the operator h as…

A: For a square matrix, the trace is the sum of the diagonal entries. Since diagonal can be obtained…

Q: Let L: R→ R be a linear operator such that L X2 )= X3-X2, then L Select one: O a. b. O d.

A: Solution of question as follows

Q: Let r E R and --(), v- (), -- (1). 3 u = V = W = 2 For which values r ER is the set {u, v, w}…

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Q: Consider ф: C" — C defined by ф(x) %3D х*. a) Kerф %3D (1,-1} b) Kerø = {i,-i} %3D с) Kerф %3D (1,1}…

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Q: Let L: R → R be a linear operator such that L X2 )=| X1 then Ker(L) spanned by: Select one: а. b.…

A: Given a linear operator L from R3 to R3. We need to find the set of vectors that spans Ker(L)

Q: Let T : R? → R² and T2 : R? → R' be linear transformations defined as follows. (:)-L (E)-) -4x1 T1…

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Q: If V = W1 e ... Wk, then there exist k linear operators E,, .., E, On V such (1) each E, is a…

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Q: Let T:R→R² be a linear operator defined by T(x,y)= (w,w,) where w, =x –-y and W, =-2r- y. %3D %3D…

A: T0Tx = T(t(x))

Q: Let g C[0, 1] be a fixed function. Consider the operator A = B(C[0, 1]) defined by (Au) (s) :=…

A: Compact Operator: A compact operator is a linear operator A:X→Y where X, Y are normed vector spaces…

Q: Conversely, if T :V V is a linear operator such that T2 = T, show that V = kerT O imT. [Hint: 3.b, v…

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Q: 20) Let f, g E F(R,R) be the functions defined by f(t) = et and g(t) = est,r ± S. Prove that f and g…

A: Answer

Q: Q NO 3 Let T: P2(R) → P2(R) be a linear operator such that T(1) = 0, T(1+ x) = 1 and T(1+ x²) = 2x.…

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Q: Consider the operator T: 1 given by T(r1, 22,. ) = (0, r1, 22,. ). Is T a compact operator? Explain…

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Q: Ques. 3: Check whether the operator T: R2 → R2 defined by T(x, y) = (y,x) is linear. Moreover if T…

A: A mapping T:R2→R2 is a linear transformation if and only if a) T(u)+T(v) = T(u+v) b) T(cu) = cT(u)…

Q: Let T: R3 → R³ defined as T (x,y, z) = (x,x – y, 2x – z). Then adjoint operator T* (u, v, w) is A.…

A: We have to find the adjoint operator T*.   According to our guidelines we can solve only 1 question…

Q: Let L:R→ R be a linear operator such that L X2 )=x1, then Im(L) spanned by: -> X3 Select one: a. 一 O…

A: The given linear operator is: L:ℝ3→ℝ3 such that Lx1x2x3=x1x1x1

Q: Suppose L: R → R° be defined by u, +17 Lu2 2u2 %3D Uz Uz Determine whether L is a linear operator or…

A: According to question given :L:R3→R3Lu1u2u3=u1+12u2u3We have to determine whether L is a linear…

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.