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(Solved): Exercise 1. For each of the bilinear forms on Rn below, find the matrix of the bilinear form with r ...
Exercise 1. For each of the bilinear forms on below, find the matrix of the bilinear form with respect to the standard basis, determine if the bilinear form is symmetric, and determine if it is a scalar product in . (a) (b) , given by for all and in (c) , given by for all and in
Expert Answer
(a) The matrix of f with respect to the standard basis of $\mathbb R^2$ is $$[f] = \begin{pmatrix}1 & -1 \\ 1 & -1 \end{pmatrix}.$$ The bilinear form f is not symmetric since $f\left(\begin{pmatrix}1 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 1\end{pmatr...