Is T: M22 --> R a Linear Transformation?

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Linear Transormation?

Homework Statement



Is function T a linear transformation for T: M22 --> R by

T [a b]
[c d] ( i.e. T [a b ; c d] )= a + b + c + d

Homework Equations

T ( u + v) = T(u) + T(v)

T(ru) = rT(u)

The Attempt at a Solution



The matrix form is confusing me
 
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Let U be the matrix you defined and let r be an arbitrary scalar. You know how to calculate rU and you have the formula for calculating T(U), so does T(rU) = rT(U)?

For the next part, define another matrix similarily as you did except with different letters. Call it V. Does T(U + V) = T(U) + T(V)?
 


the first part T(rU) = rT(U) satisfies
 


and i so does the second part! Am I right?
 


If you can show it then you are done.
 

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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