# Thread: Why is this set a subspace ??

1. ## Why is this set a subspace ??

Why is the following set a subspace of $\displaystyle M_{n \times n}^R, n \geq 2$

$\displaystyle B = \{A \in M_{n \times n}^C | A_\downarrow_1 = (1-2i)A_\downarrow_2 \}$

I immediately said it is not a subspace since complex numbers aren't in real numbers. It's the other way around. But I am wrong apparently...

2. Originally Posted by jayshizwiz
Why is the following set a subspace of $\displaystyle M_{n \times n}^R, n \geq 2$

$\displaystyle B = \{A \in M_{n \times n}^C | A_\downarrow_1 = (1-2i)A_\downarrow_2 \}$

I immediately said it is not a subspace since complex numbers aren't in real numbers. It's the other way around. But I am wrong apparently...
This is NOT a vector space over the real numbers but it IS a vector space over the complex numbers, Just as the original space, $\displaystyle M_{nxn}^C$ is a vector space over the complex numbers, not the real numbers.

3. This is NOT a vector space over the real numbers but it IS a vector space over the complex numbers, Just as the original space, is a vector space over the complex numbers, not the real numbers.
So is it a mistake that the question says it is a subspace over $\displaystyle M_{n \times n}^R$