Produce a basis for an eigenspace?

Questions
1

Hi, trying to use Mathematica to produce a basis for each eigenspace of a matrix A. If I run the command

Eigensystem[[A]

Will this always produce the eigenspaces and their basis?

For example,

Eigensystem[{{7, 3}, {3, -1}}]

produces

{{8, -2}, {{3, 1}, {-1, 3}}}

From what I understand, 8 is an eigenspace with the basis {3,1} and -2 is an eigenspace with the basis {-1,3}. Is this true?

2

Eigensystem[m] gives a list {values,vectors} of the eigenvalues and eigenvectors of the square matrix m.

3

Is this a math question or a question specific to Mathematica? If the latter, what aspect of the documentation for Eigensystem is causing confusion?