# 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 $$\big(\text{Re}(w) + \text{Im}(z) i, \, \text{Re}(z) + \text{Im}(w) i\big)$$.
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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