The set of all n×n matrices having trace equal to zero is a subspace W of $M_{n\times n}(F)$ . Find a basis for W. What is the dimension of W?. A good hint would be helpful, thanks.
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First, what is the dimension of $M_{n\times n}(F)$ over $F$? How many linear equations is $\operatorname{tr}(A) = 0$? Ok, I guess these are some hints to get you started. – Malkoun Oct 21 '19 at 13:59
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1As for a hint, could you answer the question if it were instead phrased as finding a subspace of $\Bbb F^{(n^2)}$ subject to the first $n$ entries adding to zero? The question is otherwise identical. The only difference is in flavor and how the elements of your space are represented, either as a square grid or as a very long tuple. – JMoravitz Oct 21 '19 at 14:00
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Although this question is already answered, I am giving some hints:
Hint: Because dimension of $M_{n \times n}(F)$ is $n^{2}$ and $\lbrace E_{ij} | 1 \le i,j \le n \rbrace$ is a basis, where $E_{ij}$ is a matrix with entry $1$ at $i^{th}$ row and $j^{th}$ column and other entries are zero. Now because trace of matrix is zero, remove those basis elements which does not satisfy this condition(element with non zero entry at diagonal will be removed) and add new $n-1$ elements (satisfying trace zero property) as dimension is $n^{2}-1$.
Manoj Kumar
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