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CSS Pure Mathematics QUESTION #4106
Question 1
A basis for the subspace of \(2\times2\) symmetric matrices that does NOT include the matrix \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\) is:
  • \(\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&0\\0&1\end{pmatrix}\right\}\)
  • \(\left\{\begin{pmatrix}1&1\\1&0\end{pmatrix},\begin{pmatrix}0&1\\1&1\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}\right\}\)✔️
  • \(\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&1\\1&0\end{pmatrix},\begin{pmatrix}0&0\\0&1\end{pmatrix}\right\}\)
  • \(\left\{\begin{pmatrix}1&0\\0&1\end{pmatrix}\right\}\)
Correct Answer Explanation
The space of \(2\times2\) symmetric matrices has dimension 3. Option 2 gives 3 linearly independent symmetric matrices: verify their determinant as coordinate vectors is nonzero. They span the space without using \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\) directly.