Dimension of 7x7 Matrices w/ Zero Trace: 48

[TMac92]

1. Homework Statement


Find the dimension of the set of 7x7 matrices with zero trace

Relevant Equations
The dimension of a standard basis matrix n x n is n^2
Zero trace = sum of diagonal elements = 0

Attempt at Solution
I started with dim(M) = n^2 where M is an nxn matrix.
Then I assume you would have to subtract the dimension of an nxn matrix with zero trace
dim(M with zero trace) = 1

So in this case where M is 7x7
The dimension of the set of all 7x7 matrices with zero trace would be 49-1 = 48?

 

[Outlined]

you are free to fill all the entries of your matrix except for one of the diagonal elements (say x_{1, 1}). Any choice of the other 6 diagonal elements will fix x_{1, 1} as the trace has to be 0. So yes you are right the dimension is 7² - 1 = 48.

easier: dim kernel + dim image = dim space

A trace is a real number so dim image = 1 so dim kernel = n² - 1

 

[mathwonk]

its the kernel of a map to the scalars. check whether that map is linear and you are on your way.