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CSS Pure Mathematics QUESTION #4090
Question 1
The dimension of the subspace \(W\) of all \(2\times2\) symmetric matrices (as a subspace of all \(2\times2\) matrices) is:
  • \(1\)
  • \(2\)
  • \(3\)✔️
  • \(4\)
Correct Answer Explanation
A general \(2\times2\) symmetric matrix is \(\begin{pmatrix}a&b\\b&c\end{pmatrix}\) with 3 free parameters. A basis for \(W\) is \(\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&1\\1&0\end{pmatrix},\begin{pmatrix}0&0\\0&1\end{pmatrix}\right\}\). Hence \(\dim(W)=3\).