If T:R2 R³ is a linear transformation and the action of T on the special vectors U and V is as given, find a formula for T(X), where X is any vector in R². 2 T(V) = -2 13 U = -3 2 ¹-8 V = 5 T(U) = -1 5 +
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- Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.Suppose T: R+→P3 is a linear transformation. Let u, v, and w be the vectors given below, and suppose that T(u) and T(v) are as given. Find T(w). Use the 'A' character to indicate an exponent, e.g. 5x^2-2x+1. -5 -5 V = -5 T(u) = -9x3+14x²-5x-12 T(v) = 25x³ –36x² -9x+4 -7 u = W = %3D T(w) = 0 %3DSuppose T: R¹→P3 is a linear transformation whose action on a basis for R4 is as follows: -3 a C = II 9x²-12x+6 T 0 0 Describe the action of T on a general vector, using x as the variable for the polynomial and a, b, c, and d as constants. Use the '^' character to indicate an exponent, e.g. ax^2-bx+c. -4x²+4x-4 T −4x³−8x²+14x−14 T 2x³-6x²+5x-2
- Let 4-[1] and W-[*] A = w= -[-1³]. Find k so that there exists a vector X whose image under the linear transformation T(x) = Axis w. Note: The image is what comes out of the transformation. k = Find k so that w is a solution of the equation Ax = 0. k =Use a rectangular coordinate system to plot u = X1 TO-[1:01] T(x) = X2 6 2 -}--[~3}}· V = 5 O A. Which graph below shows u and its image under the given transformation? AX₂ 6 B. and their images under the given transformation T. Describe geometrically what T does to each vector x in R². 6 X₁ X₁ D. X₁ LyLet T: R² R² be a linear transformation that sends the vector u = (5,2) into (2, 1) and maps v = (1, 3) into (-1,3). Use properties of a linear transformation to calculate T(-3u) = ( T(-3u +9v) = ( " " " ), T(9v) = ( )