# Linear Algebra Subspace Basis Problem

1. The set of all traceless (nxn)-matrices is a subspace sl(n) of (bold)K^(nxn). Find a basis for sl(n). What is the dimension of sl(n)?

Not sure how to go about finding the basis. I know a basis is a list of vectors that is linearly independent and spans.
and for the dimension of sl(n), is it just n^2, as it's an nxn matrix.

any help would be greatly appreciated.

Dick
Homework Helper
A traceless matrix has to satisfy exactly one extra condition relative to a general matrix. What does this tell you about it's dimension? Start by describing n^2-n matrices that have zeros down the diagonal. Then describe the matrices that have only diagonal elements but are still traceless.

why is it n^2-n matrices?
I only know that traceless is where the sum of the diagonals is zero.
and I also have the following information from another problem:

the set sl(n) of all traceless (nxn)-matrices form a subspace of (bold)K^(nxn):
the zero matrix belongs to sl(n) so sl(n) is not empty.
tr(cA+B) = tr(cA) + tr(B)
tr(cA+B) = c tr(A) + tr(B)
tr(cA+B) = c (0) + 0 = 0
and cA+B (epsilon) sl(n) which means sl(n) is a subspace.

Dick