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Exercise 13 from Roman's book "Advanced Linear Algebra" (page 107).

The author gives us a vector space $V$ with $$V=S_{1}\oplus T_{1}=S_{2}\oplus T_{2}$$ and asks us to prove that if $S_{1}$ and $S_{2}$ have finite codimension in $V$, then so does $\dim V/ (S_{1}\cap S_{2})$ and $$\dim V/ (S_{1}\cap S_{2})\leq \dim T_{1}+\dim T_{2}.$$

I did a lot of work but I didn't get anywhere. Thanks for your help.

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Let $p_1\colon V \to T_1$ and $p_2\colon V \to T_2$ be the projections coming with the direct sums. What are the kernels of these maps? If you define a map \[ V \to T_1 \times T_2 \] by sending $x$ to $(p_1(x), p_2(x))$, then what is the kernel of this?

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