Discussion
Let the linear transformation \(T : F^{2} \to F^{3}\) be defined by \(T(x_{1}, x_{2}) = (x_{1}, x_{1} + x_{2}, x_{2})\). Then the nullity of \(T\) is
Answer: A
The nullity of \(T\) is 0.
The kernel consists of the vectors with \(T(x_{1}, x_{2}) = (0, 0, 0)\): the first and third coordinates force \(x_{1} = 0\) and \(x_{2} = 0\). So the kernel is \(\{0\}\) and the nullity is 0.
By rank-nullity, \(\dim F^{2} = 2 = \text{rank} + \text{nullity}\), and the images \(T(1, 0) = (1, 1, 0)\) and \(T(0, 1) = (0, 1, 1)\) are independent, so the rank is 2, confirming nullity 0. The other options confuse nullity with the dimension of the codomain (3), of the domain (2), or with the codimension of the image (1).
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