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Which are Invertible

Which of the following is/are invertible linear transformations?

  1. T1:R3R3T_1: \mathbb{R}^3 \to \mathbb{R}^3 is the transformation that takes (x,y,z)(x, \, y, \, z) to (xy,yz,zx)(x - y,\, y - z,\, z - x).
  2. T2:C2C2T_2: \mathbb{C}^2 \to \mathbb{C}^2 is the transformation that takes (w,z)(w, \, z) to (Re(w)+Im(z)i,Re(z)+Im(w)i)\big(\text{Re}(w) + \text{Im}(z) i, \, \text{Re}(z) + \text{Im}(w) i\big).
  3. VV is the vector space of all sequences of real numbers (vector addition creates a new sequence from the component-wise sums of the previous two). T3:VVT_3: V \to V is the "right shift" transformation that takes a sequence {an}n0\{a_n\}_{n \ge 0} and returns a sequence {bn}n0\{b_n\}_{n \ge 0} satisfying b0=0b_0 = 0 and bn=an1b_n = a_{n - 1} for all n1n \ge 1.

Assume C2\mathbb{C}^2 is a vector space over the complex numbers.


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