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# orthogonal direct sum

Let $(V_{1},B_{1})$ and $(V_{2},B_{2})$ be two vector spaces, each equipped with a symmetric bilinear form. Form the direct sum^{} of the two vector spaces $V:=V_{1}\oplus V_{2}$. Next define a symmetric bilinear form $B$ on $V$ by

$B((u_{1},u_{2}),(v_{1},v_{2})):=B_{1}(u_{1},v_{1})+B_{2}(u_{2},v_{2}),$ |

where $u_{1},v_{1}\in V_{1}$ and $u_{2},v_{2}\in V_{2}$. Since $B((u_{1},0),(u_{2},0))=B_{1}(u_{1},u_{2})$, we see that $B=B_{1}$ when the domain of $B$ is restricted to $V_{1}$. Therefore, $V_{1}$ can be viewed as a subspace^{} of $V$ with respect to $B$. The same holds for $V_{2}$.

Now suppose $(u,0)\in V_{1}$ and $(0,v)\in V_{2}$ are two arbitrary vectors. Then $B((u,0),(0,v))=B_{1}(u,0)+B_{2}(0,v)=0+0=0$. In other words, $V_{1}$ and $V_{2}$ are “orthogonal^{}” to one another with respect to $B$.

From the above discussion, we say that $(V,B)$ is the *orthogonal direct sum* of $(V_{1},B_{1})$ and $(V_{2},B_{2})$. Clearly the above construction is unique (up to linear isomorphisms respecting the bilinear forms^{}). As vectors from $V_{1}$ and $V_{2}$ can be seen as being “perpendicular^{}” to each other, we appropriately write $V$ as

$V_{1}\bot V_{2}.$ |

Orthogonal Direct Sums of Quadratic Spaces. Since a symmetric^{} biliner form induces a quadratic form (on the same space), we can speak of orthogonal direct sums of quadratic spaces. If $(V_{1},Q_{1})$ and $(V_{2},Q_{2})$ are two quadratic spaces, then the orthogonal direct sum of $V_{1}$ and $V_{2}$ is the direct sum of $V_{1}$ and $V_{2}$ with the corresponding quadratic form defined by

$Q((u,v)):=Q_{1}(u)+Q_{2}(v).$ |

It may be shown that any $n$-dimensional quadratic space $(V,Q)$ is an orthogonal direct sum of $n$ one-dimensional quadratic subspaces. The quadratic form associated with a one-dimensional quadratic space is nothing more than $ax^{2}$ (the form is uniquely determined by the single coefficient $a$), and the space associated with this form is generally written as $\langle a\rangle$. A finite dimensional quadratic space $V$ is commonly written as

$\langle a_{1}\rangle\bot\cdots\bot\langle a_{n}\rangle,\mbox{ or simply }% \langle a_{1},\ldots,a_{n}\rangle,$ |

where $n$ is the dimension^{} of $V$.

Remark. The orthogonal direct sum may also be defined for vector spaces associated with bilinear forms that are alternating, skew symmetric or Hermitian. The construction is similar^{} to the one discussed above.

## Mathematics Subject Classification

15A63*no label found*

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