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# Let $V$ be a vector space over a field $mathrm{F}$ such that $V=V_{1} oplus V_{2}$ for subspaces $V_{1}$ and $V_{2}$ of $V$. If $f_{1}$ and $f_{2}$ are bilinear for...

## Question

###### Let $V$ be a vector space over a field $mathrm{F}$ such that $V=V_{1} oplus V_{2}$ for subspaces $V_{1}$ and $V_{2}$ of $V$. If $f_{1}$ and $f_{2}$ are bilinear forms on $V_{1}$ and $V_{2}$, respectively, prove that there is a unique bilinear form $f$ on $V$ whose restrictions to $V_{1}$ and $V_{2}$ are $f_{1}$ and $f_{2}$, respectively.

Let $V$ be a vector space over a field $mathrm{F}$ such that $V=V_{1} oplus V_{2}$ for subspaces $V_{1}$ and $V_{2}$ of $V$. If $f_{1}$ and $f_{2}$ are bilinear forms on $V_{1}$ and $V_{2}$, respectively, prove that there is a unique bilinear form $f$ on $V$ whose restrictions to $V_{1}$ and $V_{2}$ are $f_{1}$ and $f_{2}$, respectively.

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