Consider the linear transformation T: R³ → R³ that rotates vectors 90 degrees around the line (x, y, z) = (t, 2t, 3t). (The orientation of this rotation is the one given by the direction vector w = (1, 2, 3) and the right-hand rule; we aren't going to worry too much about the orientation here.) Find the matrix representation for this transformation as follows: (a) Find the matrix representation R of a rotation around the z-axis by 90 degrees. (b) The line (t, 2t, 3t) is orthogonal to the plane x+2y+3z = 0. Find a basis (v₁, v₂) for the plane x + 2y + 3z = 0 in R³. (c) Find the matrix representation of the transformation S such that S(vi) = ei, S(v₂) = ₂, S(w) = €3 (Think carefully about how to find this!)

Consider the following. T is the clockwise rotation (0 is negative) of 60° in R2, v = (1, 2). (a) Find the standard matrix A for the linear transformation T. A = (b) Use A to find the image of the vector v. T(v) =

Let T: R2 R2 be a counterclockwise rotation through 175°. Find the matrix representation of this linear transformation. Enter decimal values accurate to at least five decimal places.

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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…