Is a Real Number Sufficient for Scalar Multiplication in a Complex Subspace?

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Homework Statement



Given W={A belonging to M2(ℂ) | A is symmetric} is a subspace of M2(ℂ) over ℂ, when showing it is closed under scalar multiplication, do I need to use a complex scalar as it is over the complex numbers, or will a real number be okay?

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The Attempt at a Solution

 
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You have to prove that it's closed under scalar multiplication using complex numbers, as the vector space M_2(\mathbb{C}) is a vector space over the complex numbers.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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