Exercise 1: For each of the following, determine whether it is a subspace of the given vector space, if it is, find a its dimension {) -- 1. = 0CR³ 2. {A € R²x2 | A is upper triangular } C R2x2 3. {A € R3x3 | A is invertible } C R³×3

Q: Q4) Solve the boundary value problem y - 3y' + 2y 0 with y'(1) = 1, y(3) = 2e* – e? h = 0.5 %3D %3D

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Q: Given the following function 1, -2t + 1, 3t, 0st<2 2<t< 3 3 <t< 5 t2 5 f(t) = t- 1, (a) Express f…

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Q: 1. Using analytical method to solve the given differential equation. A steady-state heat balance for…

A: Steady state temperature

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Q: Express the following complex number in its polar form.

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Q: Let [a] be a non-zero element of Z43. w| 42. When is an element of Z43 said to be a primitive root?

A: This question is based on primitive roots. We recall the definition and give couple of examples.

Q: 1. Prove that U, W are subspaces of V 2. Find dim(U), dim(W) 3. Prove that U +W = V {() xercise 4:…

A: Subspace of a vector space

Q: Solve for the laplace transform using the formula. F(t) = e-2t-5 G(t) = t5 - e-t + 5(t – 1)2

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Q: 2. Let L={x,y,z}with the topology r={L,ø,{x},{x,y},{z},{x,z}}. Verify whether L is normal or not.…

A: The given problem is related with topology. Given that L = x, y, z with the topology τ = L, ϕ, x, x,…

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A: The volume of a square pyramid is calculated using the formula: V=13b2h, where b is the length of…

Q: 4. Let X= {a,b,c} with the topology r ={X,ø,{a},{b},{c},}{a,b},{a,c},{b,c}}. Verify whether X is…

A: Given data X=a,b,c with the topology,  τ=X,ϕ,a,b,c,a,b,a,c,b,c

Q: Ex.2. Let t: R3 → R³ be a linear transformation such that, t (a, b, c) = (3a + c, - 2a + b, - a+ 2b+…

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Q: I. Write T if the given statement is true with respect to the underlined term(s); otherwise, write…

A: (6) Let A = { x ∈ R |  x2 = 64}, then  |P(A)| = 16 False A = {-8,8}, then |A| = 2 So, |P(A)| = 22 =…

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Q: 1 5 B = 3 -1 -2 -2 ? Why? 3 2

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Q: Prove that the sum of the angles of an elliptical triangle is more than two right angles.

A: Solution:-

Q: III. Do as indicated. 1. Use Venn diagrams to exhibit A – (BnC) = (A – B) U (A – C). %3D |

A: The set A∪B means all the elements that are in set A or in set B The set A∩B means all the elements…

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A: We have to find the correct option.

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Q: (a) Find the first order partial derivatives, and ay' YOO of z = x'y+x In y + y°e* F OL LECHMoroGA…

A: Solution :-

Q: 4. Using laws of set theory aided by two-column proof, show that AU (B – C) = (AU B) – (C – A). %3D

A: Solution As we know A-B = A ∩ Bc

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Q: Solve using the method of variation of parameters: y" + 9y = 12 sec 3x. %3D

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Q: Let K be a commutative ring with identity, and let A and  B be a × a matrices over K. Then…

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A: System of Linear equations

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A: Use the following formulae, to obtain the solution.…

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A: We can solve the above problem within the following manner. Case 1: Step1: Start with person and…

Q: Q4/ Find the abs.max cand abs, min Valuse of the Runctia Kcx, y) = I+xy-3x on the closed regen…

A: We have f(x, y) = 1+xy-3x ∂f∂x = y-3, ∂f∂x = 0, y-3 = 0, y = 3 ∂f∂y = x, ∂f∂y = 0, x=0 So one…

Q: 1 1 0 3 9. 1 1 1-2 3 3 2 -1

A: Solution:-

Q: Express the following complex number in its polar form. 2+5i 6.- -4-7i

A: Solution :-

Q: L Given the figures below, do as indicated. Set up the formula only (using any method) to compute…

A: The volume of the solid of revolution for the curve y=f(x) about x-axis in the interval [a,b] is…

Q: II. Let U = R I = {x € R|x° – 14x³ + 45x = ( J = {x €R|x° 2}. Find the following. All infinite sets…

A: I={-3,-5,0,5,3} We have I={x∈ℝ|x5-14x3+45x=0} Solve for x in  x5-14x3+45x=0 We get I={-3,-5,0,5,3}…

Q: 8. Find the volume of the solid generated by revolving the region bounded by the graphs of y = x² +…

A: 1st we will draw region bounded and then rotate it about given line to form a solid of revolution .…

Q: find the minimum value of

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Q: 7. Let D = {x € Q|x² = 5}. Then, D is singleton. 8. Let H = {x € R| -1<x<1}. Then, H is finite. %3D…

A: Note:- As per our guidelines, we can answer the  three sub-parts of this problem as exactly three…

Q: 5.19 Consider the mapping T: L'(0,2) L'(0, 2), (Tf)(x):= f(t)dt. (i) Show that T indeed maps L'(0,…

A: We will use the basic knowledge of functional analysis to answer both the parts of this question…

Q: 5. Let U, W be subspaces of a vector space V such that (v, ) is a basis for U and (v2, es, 4) is a…

A: We shall use the formula dim(U+W)= dim U + dim W - dim (U \cap W) to solve this problem. The detail…

Q: I. Write T if the given statement is true with respect to the underlined term(s); otherwise, write…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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Q: 2) Using Stokes theorem, evaluate pvdr where v= 2y'i– 2x° j+÷k_and the C is the curve of C…

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Q: If C is the constant of integation, and g(0) =1 the C=___ and g(-2) = ____

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Q: For any two vectors ū and i, ||ū + u|| = ||ū|| +||7||-

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Q: Consider the function y = 4x + 3 between the limits of x = O and x = 7. а) Find the arclength L of…

A: We will use the following formulas to solve the both parts arc length of y=f(x) between x=a and a=b…