Task 1 Verify that a given mapping L is linear if: L(x,y,z) =(3*, y, 2z),x,y,z E R. If L is a mapping then determine a) Im L b) Ker L

Q: Question 2. Let T : R2 → R³ and S : R³ –→R be linear maps. Are the following compositions defined?…

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Q: 2. Let (X,r) be a topological space. Where X {a,b.c,d,e} and…

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Q: ü = [1,1, 0] and i = [-1,1,0]. Find the cross product

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Q: Question 7. Let (X,r) be a topological space and A, BCX, Determine whether A-B2A-B or not.

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Q: Tx) and (X, Ty) be two topological spaces. Let X-{a, b, c, d}. Tx-{4, X, {a}, {a, b}, {a, b, c}} Y…

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Q: Let v be represented by the directed line segment from (0, 0) to (3, 2), and let u be represented by…

A: Given that, v is a line segment from (0, 0) to (3, 2) So, v=(3,2) - (0,0)= (3,2)   Given that, u is…

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

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Q: 6. Is the set S {(2, 3, 1), (1, 2, 1), (-2, 3, 3)} C R³ linearly %3D independent or dependent?…

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Q: Thm. Let v be an inner product Space then and HyE V жує ||≤|12||· 1191/ further equility is hold iff…

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Q: Thm. Let v be an V be an inner product Space V and then жує | 13 11211. 11111 further equility is…

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Q: 10. Determine which of the following maps are linear maps. If the map is linear give the matrix…

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Q: Consider the points P = (1,2,0), Q = (0,1,2), and R=(2,0,1). Let A be an arbitrary point in…

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Q: Let u = (a, b, c) and v = along v is (2, 3, 4). If u is parallel to (1, 0, –1) and the projection of…

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Q: Q4. Let (X,T) be a topological space where X = {m,n, q, r} and T = {4, X, {q}, {r}, {q,r}, {m, q,…

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Q: 2. Which one(s) of the following mappings is(are) a linear mapping? Give the matrix of it(them).…

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Q: Problem 2 Determine which of the followings are linear maps? • F: R2 → R2 given by F(x, y) = (2x +…

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Q: Solve number 1

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Q: Verify that {u₁,u₂} is an orthogonal set, and then find the orthogonal projection of y onto…

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Q: Determine the values of y such that D(9,3,-2), E(3,4,7) and F(21,y,-20) are collinear

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Q: Let X = {a, b,c, d},T= {o, X, {d}, {a,b}, {a, b, d}} and Y = (m,n, p,g},F = {@,Y, {p}, (m,n}, {m,n,…

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Q: Q. Let (X, B(R), µ) be a Borel measure space. If An B = 0, BnC = 4 µ(ANC) = 0 Determine u(AUBUC).

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Q: Q4. Consider the topological space (X,T), where X = {a, b, c, d) and T = {o, X, {a}, {b}, {c},…

A: Given: Q4. a)  X, T is a topological space, where X=a, b, c, d and T=∅, X, a, b, c, {a, b}, {a, c},…

Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

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Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Q4. Let X = {m,n, pP. 9}, T = {6. X, {p}, {m, n}, {m,n.p}} and Y = {a, b, c, d}. F = {ó, X, {d}. {a,…

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Q: Task 1 Verify that a given mapping L is linear if: 3 L (х,у,2, w) — (x — Зw, Зу — wtz),x, у, z, wER…

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Q: Consider the map T: R³ → R2 given by Determine the Image of T m(T) = span {(0,3),(1,0)} Im(T) =…

A: The map T:ℝ3→ℝ2 given by Tx1, x2, x3=x2-2x3, 3x1+x3 To find the image of T. Ifv1, v2, ...vn in the…

Q: Thm. Let V be an inner product Space and ждек then | 1 ≤ 11all. 1171/ further equility is hold iff…

A: Using the properties of inner product And  aā=|a|2 , for any complex scalar  ||cx||=|c| ||x||, for…

Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)CA -B or not.

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Q: Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

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Q: Describe each line. a) [x, y] = [2, -5] + t[0, 1] b) [x, y] = [3, 7] + [2, 0] c) [x, y, z] = [-1, 3,…

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Q: Q. Let X and Y = {r, s,p,q}, F = Given the mapping f: (X,T) f(0) = r, f(1) = s, f(2) = q, f(3) =…

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Q: (d) V = P₂ (R), W = R³, T(p) = (1,p(1), p(2)), p = P₂ (R); (e) V = P₂ (R), W = R, T(p) = (p(1),…

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Q: W and V are linear spaces with dimension 2, the base of each of which is (e,, ez). If the linear…

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Q: W and V are linear spaces with dimension 2, the base of each of which is e1, e2). If the linear…

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