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CSS Pure Mathematics QUESTION #4118
Question 1
Which IS a valid basis for the subspace of symmetric $2 \times 2$ matrices?
  • $\left\{\begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}\right\}$
  • $\left\{\begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \ 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\right\}$
  • $\left\{\begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}\right\}$✔️
  • $\left\{\begin{pmatrix} 1 & 1 \ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \ 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\right\}$
Correct Answer Explanation
Option C gives standard basis for symmetric matrices. Option A has only 2 matrices (dimension 3 needed). Option B matrices are linearly dependent. Option D contains non-symmetric matrices.