1. Prove that there does not exist a linear map from Cô to C² whose null space equals {(x1, x2, x3, X4, X5, X6) E C° : xı – x2 = 0, x3 = -x4 = x5} .

Q: C: 1. Is there a linear map from R2 to R³ such that T(1, 3) = (0, 0, 0) and T(-2,0) = (0, 0, 0) and…

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Q: 1. Justify your answers below. (a) True or False : The function T :R → R defined by T(r) = x² is a…

A: For (a), We are given the statement,  The function T : R --> R defined by T(x) = x2 is a linear…

Q: Consider the the linear maps ƒ : R² → R² and g : R² → R² given by f((x, y)) = (x − y, 2 + y) and…

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Q: 12. A linear map T: P (R) → R² satisfies T(x) = (1,-1) and T(1) (1, 2). Find a formula for T(a1x +…

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Q: Let f: R² → R be defined by f((x, y)) = 8x +9y. Is f a linear transformation? a. f((x1, y₁) + (x2,…

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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: 1. Let T: R3 -> IR3 the linear map defined by T (x, y, z) = (y + 2, x + z, x + y). Tis she…

A: We will find out the required inverse linear map if given map is invertible.

Q: (b)Let A = {1,2,3}, B = {a, b, c}, and C = {x,y, z}. Consider the following %3D relations R and S…

A: Given R=1,b, 2,a, 2,cS=a,y, b,x, c,y, c,z

Q: 4. Let L: R² → R? be a linear operator. If L((1, 2)") = (-2, 3)" and L((1, – 1)") = (5, 2)" find the…

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Q: The linear operator T : R³ → R³ is defined by T(x₁, x2, 3) = (W₁, W2, W3), where w₁ = 2x₁ + 4x2 +…

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Q: → R be defined by f((x, y)) = −4x – 2y. Is ƒ a linear transformation? Letf: R². a. f((x₁, y₁) + (x₂,…

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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: Let L: C[0, 1] → C[0, 1] be a linear map. If L(1) =x, L(x) =x^2, then what is L(3x + 2)?

A: Given that  be a linear map and .To find the value of .

Q: Find the range of the linear map T from R³ to R such that T(x, y, z) = x+3y-z. What can we say of…

A: The range of a linear transformation T:V to W  is the subset of W . The kernel of T is the subset of…

Q: Determine whether each of the following maps from R? to R³ is linear. Prove your answers. (a) F((x,…

A: Let V(F) and V'(F) be vector spaces.  T: V(F)→V'(F) is said to be linear transformation if…

Q: (a) Construct a linear map L(z)=az+b such that L(1)=i and L(i)=1 (b) Is the map you constructed in…

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Q: Consider the map Xn+1 = aX, (1 – X;;), %3D where a > 1. (i) Find the fixed points of the map. (ii)…

A: The function f(x) is a rule that associates or fixes a number to the particular point x. The Taylor…