Which are Invertible

Algebra Level 5

Which of the following is/are invertible linear transformations?

  1. \(T_1: \mathbb{R}^3 \to \mathbb{R}^3\) is the transformation that takes \((x, \, y, \, z)\) to \((x - y,\, y - z,\, z - x)\).
  2. \(T_2: \mathbb{C}^2 \to \mathbb{C}^2\) is the transformation that takes \((w, \, z)\) to \((\text{Re}(w) + \text{Im}(z) i, \, \text{Re}(z) + \text{Im}(w) i)\).
  3. \(V\) 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). \(T_3: V \to V\) is the "right shift" transformation that takes a sequence \(\{a_n\}_{n \ge 0}\) and returns a sequence \({b_n}_{n \ge 0}\) satisfying \(b_0 = 0\) and \(b_n = a_{n - 1}\) for all \(n \ge 1\).

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