1. Justify your answers below. (a) True or False : The function T :R → R defined by T(r) = x² is a linear map. False : If B = {() . (1)} and B' = -{()- ()}- (b) True or are basis for R² and T:R? → R² is any linear map, then T# ("() "()) (c) Let T : R$ → R° and T(F) = Ai where A is the standard matrix. Supposing A is invertible, compute the rank of T. %3D

Q: 10. Define a function T: P(R) → P(R) by T(ƒ) = f' Prove that T is linear and onto, but not…

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Q: Let X {1,2, 3} and Y {3,4}. Define f : X → Y as follows: = f(2) = 3 and f(3) = 4. Draw an arrow…

A: We will draw the arrow diagram for function mapping from set A to set B.

Q: 19. Determine whether the function R: PA(R) Ps(R) defined by R(p(x)) = (x- 2)p(x) is linear.

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Q: 21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁=1, x₂ = 16, x3 = 3, x4={, x3 =…

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Q: 1. (a) Let r(t) =. Sketch the path r(t) for t between -2 and 2, showing the direction of increasing…

A: Given curve : r(t)=et, e-t

Q: Example 7. Show that T: E2 → E2, defined by is not linear T((x₁, x₂)) = (x₁ + x2, x1 - x₂ +1)

A: as we now for a linear map T(01 ) = 02  where 01 is zero of domain and 02 is zero of codomain

Q: 4. Show that a sublinear functional p satisfies p(0) = 0 and p(-x) ≥-p(x).

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Q: Q.4. Let T is a transformation from R' into R’ defined by: T(x;,*;, X; ) = (2.x, –x,, 2x; – 4x,, x;…

A: 1.)Linear Transformation:                Let U and V be two vector spaces over the same field F.…

Q: Suppose a 4D space exists for variables a, b, c, and d, where a is a function of b, c, and d, then,…

A: Given: A 4D space with variables a, b, c and d. Variable a is function of b, c and d. a=fb, c ,d

Q: 1. The map T : Mat2,2(R) → Mat2,2(R) given by a +c d a b T c d а — d

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Q: (G) G)- The projection of onto the line through zero 1 is and а. (;) () " () b. C. d.

A: Let u→=11, and the line l through zero v→=00, and w→=1-1. Need to find the projection of u→ onto the…

Q: 2 1-Find the geometric location of the relation z – 3 -ži = v2. 3

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Q: 12. A linear map T: P (R) → R² satisfies T(x) = (1,-1) and T(1) (1, 2). Find a formula for T(a1x +…

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Q: 29. Consider the function f: R² →R: (x,y) → f(x,y) = g( Determine the gradient of f. 1 1 2x - 2y g(1…

A: Please check the next step for details.

Q: 1. Determine the linear dependence of the following functions: fi(t) = 2t – 3, f2(t) = t² + 1, f3(t)…

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Q: Find a path c(t) that traces the line y = 2x + 1 from (1, 3) to (3, 7) for 0 ≤ t ≤ 1.

A: Given:

Q: A function is defined as f(x,y) = 25xy + 3x³ - 7x² sin(y) aj ay a2f (ii) Find and 1325 3²x (iii)…

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Q: 2. (§1.2 Exercise 4) For each of the following functions, sketch the contour lines on xy-plane…

A: According to the Bartleby guidelines, we are supposed to answer only the first three subparts of a…

Q: dependent.

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Q: 1. Sketch the graph of the domain of the function f (x, y)= sq x / x−y​ 2. Compute the equation of…

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Q: F . dr, (c) Let f(x, y, z) = x³ + 2yª + 325 and F consists of three line segments from (1,0, 0) to…

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Q: For each of the following, determine (with argument) whether the given function T is linear. 1. The…

A: To identify which of the given function is linear transformation.

Q: 1. This two part question asks you to determine whether a given map represents a function. bogsta…

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Q: 5. Determine whether the following transformations are linear. For those that are linear, determine…

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Q: Let L: C[0, 1] → C[0, 1] be a linear map. If L(1) =x, L(x) =x^2, then what is L(3x + 2)?

A: Given that  be a linear map and .To find the value of .

Q: 6. Let f be a function such that f(1) = 1, ƒ′(1) = 4, ƒ”(1) = −2 and ² ƒ""(t)(2 – t)²dt = 3. Find…

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Q: 7. Consider the linear map f: R² R² determined by the following matrix cos(0)-sin(0) sin(0) cos(0)…

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Q: Problem 8. Determine which of the following are linear maps. Prove your answers. (a) T₁ : R² → R2…

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Q: Problem 2. Now consider the relation D' on the domain z+ where rD'y if and only if r|y. Use the…

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Q: 20. Determine which of the following set of functions is(are) linearly independent on (-∞, ∞).…

A: Since you have asked multiple questions as per company guidelines we will solve the first question…

Q: Let u be a root of ƒ = t³ − t² +t+ 2 € Q[t] and K : = (a) Show that f = mo(u). (b) Express…

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Q: Let f(x,y)=x²y-6y²-3x². Determine the relative extrema and saddle points of f, if any.

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Q: ? ? ? 1. f(t) = 4t² + 28t, g(t) = 4t² - 28t 2. f(0) cos(30), = g(0) = -8 cos³ (0) + 6 cos (0) 3.…

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Q: Let A = {1,2,3}, let S = P(A) and let RC S x S be a relation defined by the rule (X,Y) E R…

A: Given .Let  and  be a relation defined by the following rule..

Q: 3. Describe the map T* if T = L(C³) is defined by T(x1, x2, x3) = (0,x1, x2).

A: Sure, so let's break it down. The map (T∗) here is a transformation that takes a point…

Q: (b) Show that the set {V1, V2, V3, V4, V5} = {1+x, 1-x, x² + x³, 1-x¹, x² + x¹} is a basis for…

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