Let T/6 : R² → R² denote the linear transformation which rotates a vector in R- counterclockwise by T/6 radians. Let Tref : R² → R² be the linear transformation which across the line x1 = x2. x2 reflects a vector x = (a) Sketch a cartoon illustrating what these linear transformations do to the vector e1 = (b) Find the standard matrices of T/6 and Tref. (c) Find a nonzero vector v E R² such that Tz/6 © Tref (v) = T,ref o Ta/6(v) or explain why no such vector exists.

Write a detailed solution to the problem below and upload your answer to Gradescope. Define T : R² → R² by T(7) = T ([^]) = [ 3x1 2x2 2x2 • a) Let it = [ #] and 8 = [ ] ¹ be two vectors in R² and let c be any scalar. Prove that T is a linear transformation.

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-2 Use a rectangular coordinate system to plot u = ,v = 1, and their images under the 4 -1 given transformation T(x) = | Describe geometrically what T does to each vector in R2.

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Consider the function T : R2 → R² given by T(и, v) — (г, у) — (9и + 4v, 3и + 9u) Find the Jacobian of the transformation. Type your answer.

Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…

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Define a linear transformation T: P₂ → R² by T (p) = [] p(0) ->> p(0) °‹[a] : a is a real number} O The basis of the range of T is {t, t²} The entire set of all real numbers, R [8] The entire set R² [3] : 0 is a real number} : a and b are real numbers} Describe the range of T.