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CSS Pure Mathematics QUESTION #1292
Question 1
The rank-nullity theorem states that for a linear transformation \(T:V\to W\) where \(\dim(V)=n\):
  • rank\((T)+\)nullity\((T)=2n\)
  • rank\((T)+\)nullity\((T)=n\)✔️
  • rank\((T)=\)nullity\((T)\)
  • rank\((T)\cdot\)nullity\((T)=n\)
Correct Answer Explanation
The Rank-Nullity Theorem (Dimension Theorem): for a linear transformation \(T:V\to W\) with \(\dim(V)=n\), \[\text{rank}(T)+\text{nullity}(T)=n\] where rank\((T)=\dim(\text{Im}(T))\) and nullity\((T)=\dim(\ker(T))=\dim\) of the null space.