Consider the linear functions f : R² → R³ and g : R³ → R given by f(x1,72) = 9(41, 92: 43) = Y1 + 43 – 242- T1 - 2 (a) Write down matrices A and B such that f(x) = Ax and g(y) = By, where x" = (21 r2) and y" = (n 2 ys). (b) Say whether or not each of the following functions exist: fo g, gof, f+ 29. (c) Compute the matrix C for which (g o f)(x) = Cx.

Q: (6) Define T1,T2,T3 : R³ → R³ by T1 T3( y ) Y y – z (a) Find TT2( y). (b) Suppose that T1,T2, T|T2…

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Q: Determine: If S = {X2,1 + X2} is linearly independent or dependent.

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Q: Let x and a be vectors in R2 Define the L: R2→ R2 by L(x) = x +a Is this function a linear operator?…

A: Linear operator should satisfy L:Rn→Rn  a) Lu+v=Lu+Lv         u,v∈Rn b)Lcu)=cL(u

Q: 2. Let A € Rnxn. (A is square matrix or order n with real entries). Define f: RR by f(x) = x¹ Ax, x…

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Q: Determine whether b is in col( A) and whether w is in row(A)

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Q: Define the function T : R3 → R4 according to T (x, y, z) = (2x − 3y + z, 4x − z, 3x + 7y − 8z, −x +…

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Q: Let A € Rnxn. Prove or disprove that Ax = 0 and AA x = 0 have the same solution sets.

A: note : Since you have posted multiple questions, we will provide the solution only to the first…

Q: {r, u, d) is a set of linearly independent vectors. If x = 3r+ 5u + 2d, determine whether T = {r, u,…

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Q: Use matrix multiplication to find the dilation of (a, b) with factor k = α, where α > 1

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Q: 3 Let M₂(R) be the vector space of all 2 × 2 matrices with entries from R. Show that M₂ (R) = W₁ W₂,…

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Q: Let f(u, v) = (u - v, 1, u + v) and g(x, y, z) = xyz. Find the entry a12(i.e. the entry in the first…

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Q: R³ given by 3x₁ + 4x2x3 7x1 ·x2 + x3 5x1 + 6x2 - X3 and then calculate T(-7, 1, 9) by directly…

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Q: F(x,y, z) = 5x² + 2y² + 4xy + xz

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Q: Draw the projection of b onto a and also compute it from p = îa: -=[] a = (a) b = cose 3] sin and…

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Q: 4. In the vector space P3(R), let S = {p1(x), p2(x), p3(x)}, where P1 (x) = 63x° + 19x² – 37x – 41,…

A: (a) we see, p2(x) = -3p1(x), so S is not linearly independent. [Ans] (b) As S is not linearly…

Q: Find the standard matrices A and A' for  T = T2 ∘ T1 and T' = T1 ∘ T2. T1: R2 → R2, T1(x, y) = (x −…

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Q: Let T: R² →→ R² where Find the matrix A such that T A = ([₁]) = ^ [²] · A T ( [7¹]) = [-53] and T…

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Q: Let →a=⟨7,−5⟩ and b→=⟨1,3⟩. Compute the dot product. →a X →b=

A: Given a→ = 7,-5 and b→ = 1, 3. We have to compute the dot product a→·b→.

Q: Let f(u, v) = (u - v, 1, u + v) and g(x, y, z) = xyz. Find the entry a12 (i.e. the entry in the…

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Q: Let T be a linear operator defined on P3 by T(a0+a1x+a2x^2)=a0+a1(x-1)+a2(x-1)^2. The third row of…

A: Option (a) is correct