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CSS Pure Mathematics QUESTION #4103
Question 1
If \(T:\mathbb{R}^3\to\mathbb{R}^2\) is a linear transformation with \(\text{rank}(T)=2\), then by the Rank-Nullity Theorem, the dimension of the kernel is:
  • \(0\)
  • \(1\)✔️
  • \(2\)
  • \(3\)
Correct Answer Explanation
Rank-Nullity Theorem: \(\dim(\ker T)+\text{rank}(T)=\dim(\mathbb{R}^3)=3\). Given \(\text{rank}(T)=2\): \(\dim(\ker T)=3-2=1\).