Your 2nd problem is to determine whether the set of 2 x 2 matrices
<br />
\left[ \begin{array}{ c c }<br />
a & b \\<br />
c & d<br />
\end{array} <br />
\right]
where a + d = 0, is a subspace (of dimension 3) of the vector space of 2 x 2 matrices.
There are two parts to this problem:
- Showing that this set of matrices is a subspace.
- Finding the dimension of this subspace.
For the first part, show that:
- The 2 x 2 zero matrix belongs to this set.
- If M1 and M2 are in this set, then M1 + M2 is also in the set.
- If M is a matrix in this set, and c is a scalar (a real number), then cM is also in this set.
For the second part you have two pieces of information to work with: the equation a + d = 0, and the fact that the entries of matrices in this set are a, b, c, and d, reading across the rows.
From the equation you are given, you can get four equations:
a = -d
b = b
c = c
d = d
Equivalently, this system is:
Another way to look at this system is that the entries on the left side represent your matrix (as a vector), and the right side entries can be thought of as the sum of 3 vectors/matrices.
That is,
<br />
\left[ \begin{array}{ c }<br />
a \\<br />
b \\<br />
c \\<br />
d<br />
\end{array} <br />
\right]=
d\left[ \begin{array}{ c }<br />
-1 \\<br />
0 \\<br />
0 \\<br />
1<br />
\end{array} <br />
\right]
+
b \left[ \begin{array}{ c }<br />
0 \\<br />
1 \\<br />
0 \\<br />
0<br />
\end{array} <br />
\right]
+
c \left[ \begin{array}{ c }<br />
0 \\<br />
0 \\<br />
1 \\<br />
0<br />
\end{array} <br />
\right]<br />
