Suppose that T is a one-to-one transformation, so that an equation T (u) = T(v) always implies u = v. Show that if the set of images {T(v1), . .. , T (vp)}islinearly dependent, then {V1, ..., Vp is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set (because in this case the set of images cannot be linearly dependent).

Q: 5. T: P3R, T (a₁ + a₁x + a₂x² + a3x³) = a₁ + a₂ Answer

A: Kernel is the elements of P3 whose image under transformation T is zero in R.

Q: Let f:  V-->W be any linear transformation. What are possible solutions to f(0)?

A: Note: A Linear Transformation Maps the Zero vector to the Zero Vector.

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Q: Construct a linear fractional transformation that maps 1, -1, 2 to o, i, 0, respectively.

A: Fractional linear transformation

Q: Show that the transformation T defined by T[x1, X2) = (2x, - 3x.2, Xq + 4, 5x,) is not linear. %3D

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Q: T(x, y, z) = (x + y, z-1)

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Q: If {~v1,··· ,~vr} is linearly independent and T is a one to one linear transformation, show that…

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Q: a) T: M, → R, where M, is the vector space of n by n matrices over a field, and T (A) = det(A).

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Q: a b L:M,(R)→R’, L 2)-(a+26,b-3c,c+d)

A: First I have written the definition of Linear Transformation , just use this definition for given…

Q: ). Find T(1, 3) and 7(1, -5).

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Q: determine if the following transformation is linear or not wwww win i X -y. T:R3--> R? t (y) = %3D…

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Q: Let T: R" → R" be a linear transformation such that 7(20, +30₂) = -50, +30, and 7(30₁ +50₂) = 30,…

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Q: Example : Define a linear transformation T:R² → R² by T(2) = [: [ X1 - 2 | 1 X2 X1 Find the images…

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Q: T(xy)=Cax?+by?, cx²+dy²) 2

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Q: A ||

A: T:P2→P2 defined by  Tft=f9t+6

Q: Show that projw is a linear transformation and that (projw)² = projw.

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Q: Suppose that T is a linear transformation, with T(u,) = T(u2) =:: Find T(2u, - 3u,).

A: From the definition of linear transformation we can solve this question.

Q: Assume that T is a linear transformation and T(1,2,1)=(2,1,3), T(-1,2,3)=(1,1,3), 7(2,1,3)=(1,-1,3),…

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Q: 1. For each of the following sets S of vectors in R", find a linear transformation L : R" → R³ whose…

A: For a linear mapping T:R→S,the kernel of T is the set of elements in R which maps to zero vector in…

Q: T: P2 P2, .2 T(ax2 + bx + c) = ax² + bx +c B = {1, 1 – x, (1 - x)²} B' = {1, x , x²} v = x² – 3x +3…

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Q: When possible, an example of a linear transformation T that satisfies the condition in hint on each…

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Q: 3. Let B = Consider T : R2,2 → R²,² defined by T(A) = AB – BA. Let [o o] Bị and B2 :

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Q: I just need an example of a linear transformation V to V that is injective but not surjective

A: