course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
36 |
|
Paper 1, Section I, E |
2018 |
State the Rank-Nullity Theorem.
If $\alpha: V \rightarrow W$ and $\beta: W \rightarrow X$ are linear maps and $W$ is finite dimensional, show that
$$\operatorname{dim} \operatorname{Im}(\alpha)=\operatorname{dim} \operatorname{Im}(\beta \alpha)+\operatorname{dim}(\operatorname{Im}(\alpha) \cap \operatorname{Ker}(\beta))$$
If $\gamma: U \rightarrow V$ is another linear map, show that
$$\operatorname{dim} \operatorname{Im}(\beta \alpha)+\operatorname{dim} \operatorname{Im}(\alpha \gamma) \leqslant \operatorname{dim} \operatorname{Im}(\alpha)+\operatorname{dim} \operatorname{Im}(\beta \alpha \gamma)$$