$M_2$ is the vector space of all $2\times 2$ matrices with real entries. For what real number r is the set $\left.\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\right|a+b+c+d=r\right\}$ a subspace of $M_2$?
For what real number $r$ is the set \left.\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\right| a+b+c+d=r\right\} a subspace of $M_2$?
0
$\begingroup$
matrices
vector-spaces
-
1Peng, now that you can answer the question, you should post an answer. Later, you can accept it. – 2012-10-21
1 Answers
4
A quick solution:
Every subspace must contain zero vector, which means $\left\{\begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}|0+0+0+0=r\right\}$ is in the subspace of $M_{2}$. Hence, $r$=0.
Alternative solution:
We can use the definition of subspace to find $r$ : closed under addition and scalar multiplication.