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Is the set of 2x2 matrices with sum of diagonal entries equal zero a vector space over the real numbers under the usual matrix matrix addition and scalar matrix multiplication?

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Yes. You can check it directly or note that the set is given by

$$ \left\{ \left. \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \right| \, a_{11} + a_{22} = 0 \right\} $$

and so it is the set of solutions to a linear homogeneous equation and thus a vector space of dimension $4 - 1 = 3$.

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  • $\begingroup$ Thanks for the edit! I was wondering for a long time how to make the set theoretic brackets to match the size what they surround. $\endgroup$ – levap Nov 10 '15 at 13:07
  • $\begingroup$ but a11+a22 must be zero. Diagonal entries sum is zero $\endgroup$ – user288787 Nov 10 '15 at 20:03
  • $\begingroup$ You are right, but it doesn't matter which linear equation you write - the solution space will still be a vector space. $\endgroup$ – levap Nov 10 '15 at 21:51
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Set of all matricess of below type with a,b,c $\in \mathbb{R}$ \begin{bmatrix} a & b\\ c & -a\\ \end{bmatrix} are the $2 \times 2$ matrices of trace zero form a vector subspace of $M(2,\mathbb{R})$ with dimension 3. Since, there are three free arbitrary constants(a,b,c). Vector addition and scalar multiplication closed under usual matrix addition and scalar with matrix multiplication.

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