# Linear Transformations

1. Oct 22, 2009

### Iconate

1. The problem statement, all variables and given/known data

Is the function which rotates the xy-plane by 20 degrees is a linear transformation?

From R2 -> R2

2. Relevant equations

x = xcos$$\theta$$ + ysin$$\theta$$
y = -xsin$$\theta$$ + ycos$$\theta$$

Where $$\theta$$ = 20 degrees (or $$\pi$$/9 )

3. The attempt at a solution
Apparently the solution to this is true:

Ok so the two properties must hold

T(u + v) = T(u) + T(v)
and T(cu) = cT(u)
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Let u = (u1, u2) and v =(v1, v2)

T(u) + T(v) = u1cos$$\theta$$ + u2sin$$\theta$$, - v1sin$$\theta$$ + v2cos$$\theta$$

However this isn't equal to T(u+v) = T((u1 + v1), (u2 + v2)) which when expanded will give me more cos$$\theta$$ terms than in T(u) + T(v).

I figure im just not computing this properly, any help would be great, thanks.