8. Let $$ x_{1}, x_{2}, x_{3}, x_{4} $$ be the co-ordinates of a vector $$ X $$, and $$ y_{1}, y_{2}, y_{3}, y_{4} $$ the co-ordinates of a vector $$ Y $$. Then find the bilinear form associated with the following quadratic forms. $$ \begin{array}{lll} ext { (i) } x_{1} x_{2} & ext { (ii) } x_{1} x_{3}+x_{4}^{2} & ext { (iii) } 2 x_{1} x_{2}-x_{3} x_{4} \ ext { (iv) } x_{1}^{2}-5 x_{2} x_{3}+x_{4}^{2} & \end{array} $$

Question

8. Let $$ x_{1}, x_{2}, x_{3}, x_{4} $$ be the co-ordinates of a vector $$ X $$, and $$ y_{1}, y_{2}, y_{3}, y_{4} $$ the co-ordinates of a vector $$ Y $$. Then find the bilinear form associated with the following quadratic forms. $$ \begin{array}{lll}	ext { (i) } x_{1} x_{2} & 	ext { (ii) } x_{1} x_{3}+x_{4}^{2} & 	ext { (iii) } 2 x_{1} x_{2}-x_{3} x_{4} \ 	ext { (iv) } x_{1}^{2}-5 x_{2} x_{3}+x_{4}^{2} & \end{array} $$

Lambert Ward · Accepted Answer

The bilinear forms associated with the given quadratic forms are: 1. **(i)** $$ B(X, Y) = x_{1} y_{2} + x_{2} y_{1} $$ 2. **(ii)** $$ B(X, Y) = x_{1} y_{3} + x_{3} y_{1} + 2x_{4} y_{4} $$ 3. **(iii)** $$ B(X, Y) = 2x_{1} y_{2} + 2x_{2} y_{1} - x_{3} y_{4} - x_{4} y_{3} $$ 4. **(iv)** $$ B(X, Y) = 2x_{1} y_{1} - 5x_{2} y_{3} - 5x_{3} y_{2} + 2x_{4} y_{4} $$

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