Let f : X → Y and g: Y → Z be two maps. (a) Show that if g o f is surjective, then g must be surjective. (b) Show that if g o f is injective, then f must be injective.

Q: Prove that the mapping f: G- G defined by f(x) = x-l is an isomorphism of G onto itself.

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Q: Let S : RR be defined by S(x1, x2, 3,...) = (T2, T3, T4,...) (a) Using the definition of linear…

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Q: State whether the following is true or false: Let f:R2 R2 be a mapping such that f(1,1) = (3, -6)…

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Q: Let T: R2 - R3 be the linear map which satisfies and 3 Give the formula for T. x + 3y x + y

A: We have to find

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

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Q: Let m, n e Z4 and X + 0. a. If m < n, define an injective map f : Xm → x". b. define a bijective map…

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Q: Q4. Let X = {m, n, p, q}, T = {4, X, {p}, {m, n}, {m, n, p}} and Y = {a, b, c, d}, F = {ø, X, {d},…

A: There is a slight error in the question. X cannot be an element of F. Instead of X, it will be Y.…

Q: Suppose that T: R³ → R³ is a linear map, and ... T -0-0 T2 = Then F O O O 0- = ... [] 00 сл 2 H 23 6…

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Q: Let T = {0, 1, 2, 3} and the relation R is defined on T as follows: R = {(0, 1), (1, 2), (2, 2)}.…

A: The symmetric closure, R(s) is calculated using the formula, R(s)=R∪R-1 , where, R-1=b,a: a,b∈R .…

Q: (3) Conjugate the map f(x) = 2x² +x – 3 to a map of the form Q(x) = x² + c. Find the conjugacy map h…

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Q: 4. Consider the the linear maps f: R² R² and g: R² R² given by f((x, y)) = (x-y,2+ y) and g((x, y))=…

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Q: a linear map f: EF, the followi fImf is isomorphic to Im fT = f(- (Im f)* ≈ Im fT.

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Q: Let X be the set {1, 2, 3, 4, 5, 6}, Y the set {a, B, 7, 8}, and ƒ the map f(1) = a f(2)= a ƒ(3) = d…

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Q: Determine whether the functions fi(t) = 3t – 5, f2(t) = 3t² + 1, f3(t) = 5t² + t are linearly…

A: In this question, concept of linear dependent is applied. If there is a nontrivial linear…

Q: Let X be the set {1, 2, 3, 4, 5, 6}, Y the set {a, ß, 7, 6}, and ƒ the map f(1) = a f(2)= a f(3) = a…

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Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

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Q: 2. Let : RR and y: R+ → R be mappings with 6(x) -2e-x. (a) Find the range of y. = 1. Use your answer…

A: Here's a summary of the answers:  Solutions:(a) Find the range of γ (b) Use the definition to show…

Q: Let G be a direct product of H and k product of 2 (H) 'and z (k) then Z(G) is a direct

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Q: Define a linear transformation T : P, → R² by T(p : p(-1) + p(2) p(3) (a) Find the kernel space of…

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Q: Suppose R1 is a relation on domain X1 and R2 is a relation on domain X2 Define a new relation R…

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Q: Determine whether the functions fi(t) = 2t5, f2(t) = 2t²+1, f3(t) = 5t² + t are linearly dependent…

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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: (b) Let X=ZX (Z \ {0}). Define the relation = on X by (x, y) = (z,t) ⇒xt = yz for every (x, y),…

A: Given: Let X=ℤ×ℤ \ 0. Define the relation ≡ on X by  x,y≡z,t⇔xt=yz for every x,y, z,t∈X. To prove:…

Q: Let T : R3 → R² be a linear mapping. 2 .T Given that T and T find T

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Q: Determine if the map g : Q ! Q dened by g(x) = x/(x2+1) is one-to-one.

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Q: Consider the linear map T : C³ → C² given by T 21 23 (a) Determine the range of T. (b) Determine the…

A: Solution:-

Q: 2. Suppose that G is a lincar map such that G(2,0) = (4, 0) and G(0, 3) = (–3, 9). Find the images…

A: Given that, G(2, 0) = (4, 0) G(0, 3) = (-3, 9) G is a linear map. Then we get, (a) G(1, 0) = (2, 0)…

Q: ii. Let f: A → B and g: B → C be invertible mappings; that is, mappings such that f-¹ and g-¹ exist.…

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Q: Let A B and g: B-C be maps. Prove that if g of is one-to-one, then / is one-to-one.

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Q: f: Z→ No, x→ 2x, -2x - 1, when falls x ≥ 0 falls x < 0 مید ماس

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Q: If n&lt;m then every linear map T: IRn -&gt; IRm is one-to-one. Is this statement true or false?

A: The given statement is

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: Use the mathematical definition of concave and quasi-concave functions to prove that: 3. (a) The sum…

A: A real-valued function f {\displaystyle f} on an…

Q: is any linear map, then the foll Im f¹ = (Ker (f))º Ker (f¹) = (Im f)⁰ T0

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Q: Let M(R) denote the set of all mappings from R to R. For mappings f,g: R → R, define pointwise…

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Q: (y – 2)² Draw a contour map of z = r² + using k = 0, 1,2, 3, 4, then sketch its graph in 9 R³.

A: The graph of a three dimensional surface is represented by the equation: z=f(x,y). The graph of a…

Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

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Q: Suppose T: V --> W is a linear and surjective map and V and W are of finite dimension. Show that T…

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