4.I .6 F. 6 \mathrm{~F} \quad

Linear Mathematics | Part IB, 2002

Define the rank and nullity of a linear map between finite-dimensional vector spaces.

State the rank-nullity formula.

Let α:UV\alpha: U \rightarrow V and β:VW\beta: V \rightarrow W be linear maps. Prove that

rank(α)+rank(β)dimVrank(βα)min{rank(α),rank(β)}\operatorname{rank}(\alpha)+\operatorname{rank}(\beta)-\operatorname{dim} V \leqslant \operatorname{rank}(\beta \alpha) \leqslant \min \{\operatorname{rank}(\alpha), \operatorname{rank}(\beta)\}

Part IB

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