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I studied this subject a time ago, but now I have to answer this question. But the big problem is that my old exercises are different. I can't find a similar question on internet. Can you help me?

-- Let V the vector space of 2x2 matrices in $\mathbb{R}^{2 \times 2}$. Let W be the subspace of symmetric matrices. Show that dim W = 3, determining a basis of W.

Can you help me? Should I use kernel?

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Hint:

Rather than thinking of $2\times 2$ matrices: $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ which are symmetric, you can think of the related question of vectors $\begin{bmatrix}a\\b\\c\\d\end{bmatrix}$ if that helps you think.

Additional hint:

A symmetric $2\times 2$ matrix of the form $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ must have $b=c$

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  • $\begingroup$ Thank you. Is the solution like this? math.odu.edu/~bogacki/cgi-bin/… $\endgroup$ – user17245 Jun 18 '18 at 15:26
  • $\begingroup$ @user17245 Remember that the original question was written in terms of our space consisting of $2\times 2$ matrices... so your answer should also be written in terms of $2\times 2$ matrices. $\endgroup$ – JMoravitz Jun 18 '18 at 15:37
  • $\begingroup$ I get it...thanks. But the calculator above is correct, should I only write in 2x2 matrices? $\endgroup$ – user17245 Jun 18 '18 at 15:45
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Let $B_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}, B_2=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and $B_3=\begin{bmatrix}0&0\\0&1\end{bmatrix}$.

Then each $B_j$ is symmetric. Show that $\{B_1,B_2,B_3\}$ is linearly independent and that $W=span (\{B_1,B_2,B_3\})$.

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