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Thread: bilinear forms

  1. #1
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    bilinear forms

    the problem statement is, Prove that there is a unique bilinear form f, on Um X Vn such that, f(ui,vj)=aij where aij, i=1,...,m j=1,..n are given scalars, {u1,u2,...um} and {v1,v2,...vn} are ordered bases for Um and Vn respectively.

    i've got how to prove it is bilinear but how do i prove uniqueness?
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  2. #2
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    Quote Originally Posted by neha18 View Post
    the problem statement is, Prove that there is a unique bilinear form f, on Um X Vn such that, f(ui,vj)=aij where aij, i=1,...,m j=1,..n are given scalars, {u1,u2,...um} and {v1,v2,...vn} are ordered bases for Um and Vn respectively.

    i've got how to prove it is bilinear but how do i prove uniqueness?
    Just assume that you are given a second billinear form g that satisfies $\displaystyle g(u_i,v_j)=a_{ij}$ as well (and, therefore $\displaystyle g(\vec{u}_i,\vec{v}_j)=f(\vec{u}_i,\vec{v}_j)$ for all $\displaystyle i=1,\ldots,m; j=1,\ldots,n$) and then show that for all vectors $\displaystyle \vec{x}, \vec{y}$ it is true that $\displaystyle g(\vec{x},\vec{y})=f(\vec{x},\vec{y})$.
    You do this by replacing $\displaystyle \vec{x}$ and $\displaystyle \vec{y}$ by linear combinations of the basis vectors, $\displaystyle \vec{x}=x_1\vec{u}_1+\cdots+x_m\vec{u}_m$ and $\displaystyle \vec{y}=y_1\vec{v}_2+\cdots+y_n\vec{v}_n$ for example, and apply bilinearity...
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