# Why is this set a subspace ??

• Jul 3rd 2010, 03:08 PM
jayshizwiz
Why is this set a subspace ??
Why is the following set a subspace of $M_{n \times n}^R, n \geq 2$

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

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...
• Jul 3rd 2010, 03:49 PM
HallsofIvy
Quote:

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

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

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, $M_{nxn}^C$ is a vector space over the complex numbers, not the real numbers.
• Jul 3rd 2010, 09:10 PM
jayshizwiz
Quote:

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, http://www.mathhelpforum.com/math-he...894ae225d7.png 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 $M_{n \times n}^R$