4 Ex12. Let L:R5 → R* be defined by X y EH = W L -1 3 02 -1 5 0 0 -1 1 1 0 1 0 2 0 -1 -1 0 X y Z W Find ker L, basis for ker L, dim(ker L) and determine if L one to one. Find range L, basis for range L, dim(range I and determine if L onto.

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: 4.5 Compute the value of G dL for G 2ya, with A(1,-1, 2) and P(2, 1, 2) using the path: (a)…

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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: Exercise 2. Prove that dim(null T") = dim(null T) + dim W – dim V and dim(range T") = dim(range T),…

A: We prove dim null T*=dim null T + dim W-dim V and dim range T *=dim range T for every T L(V·W).

Q: Suppose z = x² sin y, x = s² + 3t², y = 6st. Əz Əz A. Use the chain rule to find oz and 0 as…

A: Suppose .(A) By using chain rule to find  as functions of .(B) To find the numerical values of  when…

Q: Find the exact value of F. dr using any method. b. F(x, y, z)= C is the line segment from (1, 0,…

A: The given vector field is Fx, y, z=yz, xz, xy+2z. We have to find ∫CF·dr.

Q: O {1, z+ 3, 2a, sinz In 2,1

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

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Q: 18. Show that u = [2.-5.15] is orthogonal to v = [20,–1,–3).

A: From the given problem:   u→=2,-5,15 ; v→=20,-1,-3   As we know that two vectors u→ & v→ are…

Q: Let f(x, y, z) = x²y³ + zª and x = s², y = st², and z = (a) Calculate the primary derivatives - af…

A: fx, y, z=x2y3+z4  &  x=s2, y=st2,  &  z=s2t a). ∂f∂x=∂∂xx2y3+z4=y3·2x+0=2xy3…

Q: 34. Let T: R² → R² be the function that maps each point in R² to its reflection with respect to the…

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Q: 2. Let g (x, y) = 2x² – y² . Compute 9(1, 2), g (2, 1), g (1, 1), g (–1, 1) and g (2, –1).

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Q: 3.2. Show, using the two properties that define a linear function, whether each of the following…

A: Solution: (3.2) Functions are linear: (i) fx1;x2=5x1+lne6x2 Now, fx1+y1;x2+y2=5x1+y1+lne6x2+y2…

Q: Let L: R2 R³ be a linear mapping such that Find L -19 ([¹2]) -13 absin (a) 8 dr f 8 L D-||| (D-|| =…

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Q: 17. Let T: R² → R² be a linear transformation that maps -[2] into [2] = [] into [3] fact that T is…

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Q: Let L : R → R³ be a linear transformation defined by L x2 x3 What is the range of L? t, s eR 0. teR…

A: Basis of the linear transformation

Q: 6 Find all Max., Min., saddle point for the functions below: 3 f(x,y) = x ²³ + 3xy ² - 15 x +…

A: Given: f(x,y) = x3+3xy2-15x+y3-15y

Q: A₁ Let 2 + sinh (2²) - ₁₁ = 27 where w = f (x, y, z)- 1 पुर Find aw дх Jw and Iw > zw. ay dz 3/20

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Q: If F(x, y, z) = (sin x + 2) i+2xz j+ (e-x) k and the path C is the line segment from A(1,2, 1) to…

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Q: 41. Let 1 S = %3D 2 -1 } 4 1 a. Find span(S). b. Is S linearly independent? c. Let -三 T = -1 -1 -3 4…

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Q: Ex12. Let L:R5 → Rª be defined by X y ·10 L = V W 1 1 2 0 -1 3 0 0 2 0 -1 5 - 1 - 0 X y Z V W ● a)…

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Q: 8. Show that an isometric linear operator T: H→→ H satisfies T*T= I, where I is the identity…

A: Let T : H → H  be an isometric linear operator. We need to show that , T satisfies T* T = I We know…

Q: (5) Which of the following sets are linearly independent. Show all work. set 1:[1, e, ex] or set…

A: Step 1:Solution-Step 2:Step 3: Step 4: