Browse & Filter

Show that real anti-symmetric matrices in odd dimensions are not invertible.

Find the matrix representation of the  operators \(A\) on the vector space of all  \(2\times 2\) real matrices, where
\[ A \begin{pmatrix} a & b \\c & d\end{pmatrix} =
    \begin{pmatrix}1 & 1\\1& 1 \end{pmatrix}
    \begin{pmatrix} a & b \\c & d\end{pmatrix},\]
in the basis
 \begin{equation*}
  e_1 = \begin{pmatrix}
          1 & 0\\0& 0&0
        \end{pmatrix};\quad
  e_2 = \begin{pmatrix}
         0 & 1 \\ 0 & 0
        \end{pmatrix};\quad
  e_3 = \begin{pmatrix}
         0 & 0 \\ 1 & 0
         \end{pmatrix};\quad
  e_4 = \begin{pmatrix}
          0 & 0 \\
          0 & 1
        \end{pmatrix}.
\end{equation*}

Consider the vector space \(\Pbb_5(t)\) whose elements \(p(t)\) are polynomials in \(t\) of degree less than or equal to 4.
\[ p(t) =\alpha_0 +\alpha_1t + \alpha_2 t^2 + \alpha_3 t^3 +\alpha_4t^4\]  Consider the subspace
\(\Vsc_1\) of \(\Pbb_5(t)\) consisting of polynomials which are even functions  of  \(t\). What is the dimension of \(\Vsc_1\)?  What is the vector space \(\Vsc_2\) such that \( \Pbb_5(t) =  \Vsc_1 \oplus \Vsc_2\). Give a basis in \(\Vsc_2\).